Smooth a PGFplot

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7
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I'm trying to replicate a graph I created in Desmos over here,



with the following code in Overleaf



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
 declare function=normd(x,n,p)=binom(x*n,n,p);
]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


However, the resulting graph is not as smooth (and rather jaggy):
jagged



I've tried to increase the number of samples, samples=500 and samples at=0, 0.001, ..., 0.999, 1, but both still can't replicate the granularity that Desmos produces.



Are there just some functions which are hard to be plotted through pgfplot?




pgfplots




share|improve this question




















  • You could just add smooth to the plot. However, then the plot is no longer accurate, i.e. the interpolation will cause the plot to deviate from the true values (a bit). I guess that the problem is that factorial is only defined for integers. You may want to use the Gamma function for a continuous interpolation.
    – marmot
    Sep 5 at 22:27











  • Per your suggestion, I tried adding smooth into the axis parameters, but that only results in the jagged edges to being smoothed but does not create the normalized binomial plot I'm trying to create :(
    – Sentient
    Sep 5 at 22:29










  • Yes, but my comment had a second part. ;-)
    – marmot
    Sep 5 at 23:00














up vote
7
down vote

favorite












I'm trying to replicate a graph I created in Desmos over here,



with the following code in Overleaf



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
 declare function=normd(x,n,p)=binom(x*n,n,p);
]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


However, the resulting graph is not as smooth (and rather jaggy):
jagged



I've tried to increase the number of samples, samples=500 and samples at=0, 0.001, ..., 0.999, 1, but both still can't replicate the granularity that Desmos produces.



Are there just some functions which are hard to be plotted through pgfplot?




pgfplots




share|improve this question




















  • You could just add smooth to the plot. However, then the plot is no longer accurate, i.e. the interpolation will cause the plot to deviate from the true values (a bit). I guess that the problem is that factorial is only defined for integers. You may want to use the Gamma function for a continuous interpolation.
    – marmot
    Sep 5 at 22:27











  • Per your suggestion, I tried adding smooth into the axis parameters, but that only results in the jagged edges to being smoothed but does not create the normalized binomial plot I'm trying to create :(
    – Sentient
    Sep 5 at 22:29










  • Yes, but my comment had a second part. ;-)
    – marmot
    Sep 5 at 23:00












up vote
7
down vote

favorite









up vote
7
down vote

favorite











I'm trying to replicate a graph I created in Desmos over here,



with the following code in Overleaf



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
 declare function=normd(x,n,p)=binom(x*n,n,p);
]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


However, the resulting graph is not as smooth (and rather jaggy):
jagged



I've tried to increase the number of samples, samples=500 and samples at=0, 0.001, ..., 0.999, 1, but both still can't replicate the granularity that Desmos produces.



Are there just some functions which are hard to be plotted through pgfplot?




pgfplots




share|improve this question












I'm trying to replicate a graph I created in Desmos over here,



with the following code in Overleaf



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
 declare function=normd(x,n,p)=binom(x*n,n,p);
]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


However, the resulting graph is not as smooth (and rather jaggy):
jagged



I've tried to increase the number of samples, samples=500 and samples at=0, 0.001, ..., 0.999, 1, but both still can't replicate the granularity that Desmos produces.



Are there just some functions which are hard to be plotted through pgfplot?





pgfplots





share|improve this question











share|improve this question




share|improve this question










asked Sep 5 at 22:16









Sentient

17014




17014











  • You could just add smooth to the plot. However, then the plot is no longer accurate, i.e. the interpolation will cause the plot to deviate from the true values (a bit). I guess that the problem is that factorial is only defined for integers. You may want to use the Gamma function for a continuous interpolation.
    – marmot
    Sep 5 at 22:27











  • Per your suggestion, I tried adding smooth into the axis parameters, but that only results in the jagged edges to being smoothed but does not create the normalized binomial plot I'm trying to create :(
    – Sentient
    Sep 5 at 22:29










  • Yes, but my comment had a second part. ;-)
    – marmot
    Sep 5 at 23:00
















  • You could just add smooth to the plot. However, then the plot is no longer accurate, i.e. the interpolation will cause the plot to deviate from the true values (a bit). I guess that the problem is that factorial is only defined for integers. You may want to use the Gamma function for a continuous interpolation.
    – marmot
    Sep 5 at 22:27











  • Per your suggestion, I tried adding smooth into the axis parameters, but that only results in the jagged edges to being smoothed but does not create the normalized binomial plot I'm trying to create :(
    – Sentient
    Sep 5 at 22:29










  • Yes, but my comment had a second part. ;-)
    – marmot
    Sep 5 at 23:00















You could just add smooth to the plot. However, then the plot is no longer accurate, i.e. the interpolation will cause the plot to deviate from the true values (a bit). I guess that the problem is that factorial is only defined for integers. You may want to use the Gamma function for a continuous interpolation.
– marmot
Sep 5 at 22:27





You could just add smooth to the plot. However, then the plot is no longer accurate, i.e. the interpolation will cause the plot to deviate from the true values (a bit). I guess that the problem is that factorial is only defined for integers. You may want to use the Gamma function for a continuous interpolation.
– marmot
Sep 5 at 22:27













Per your suggestion, I tried adding smooth into the axis parameters, but that only results in the jagged edges to being smoothed but does not create the normalized binomial plot I'm trying to create :(
– Sentient
Sep 5 at 22:29




Per your suggestion, I tried adding smooth into the axis parameters, but that only results in the jagged edges to being smoothed but does not create the normalized binomial plot I'm trying to create :(
– Sentient
Sep 5 at 22:29












Yes, but my comment had a second part. ;-)
– marmot
Sep 5 at 23:00




Yes, but my comment had a second part. ;-)
– marmot
Sep 5 at 23:00










2 Answers
2






active

oldest

votes

















up vote
9
down vote



accepted










I get a very different result from Ruixi if I use the well-known relation Gamma(n+1)=n! to provide a continuous version of factorial. However, the result seems to match your Desmos curve very accurately, so I think it is correct. The definition of the Gamma function is from this answer, and I used it to provide a continuous generalization of factorial here, when encountering the same problem.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[% gamma definition from https://tex.stackexchange.com/a/120449/121799
 declare function=gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[blue, domain=0:1,samples=100,smooth]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



CROSS CHECKS: I first check that the Gamma function has been implemented correctly and then draw your contour with it.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=1:10,samples=10,only marks,mark=*]x!;
 addplot[blue, domain=1:10]gamma(x+1);
endaxis
endtikzpicture


begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
 addplot[blue, domain=0:1]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



As you see, the smoothed out version of your plot is very different from the one with factorial in, but factorial is per se not defined for non-integers.






share|improve this answer


















  • 1




    Oh, amazingly it looks like this would reproduce the Desmos (whatever that is) curve.
    – marmot
    Sep 5 at 22:47










  • Oof, I probably would've never though about using the Gamma function to smooth it out. Thanks!
    – Sentient
    Sep 5 at 22:52










  • @Sentient This is a standard trick which (essentially) gave t'Hooft a Nobel prize. ;-) You need it in order to compute loop diagrams in 4-epsilon dimensions, where epsilon is a real number.
    – marmot
    Sep 5 at 22:57

















up vote
6
down vote













Added: Please note that the following discussions are based on my background in probability theory. In your example, binomial probabilities are defined only when x*n are integers; that is, these probabilities are defined only when x is equal to one of the following 16 numbers



0, 1/15, 2/15, ... , 14/15, 15/15


Thus, in order to smoothly extend the graph, you have many options. These extensions include but are not limited to




  1. Analytical continuation via the gamma function. But, you should be aware of the fact that




    TeX is no Mathematica or MATLAB or Maple




    and therefore the gamma function is not available on the fly. So, @marmot provided an excellent answer in which the gamma function is hardcoded using its asymptotic expansions. Because it uses asymptotic expansions, there could be unexpected computation errors (but only if you choose extreme domain to be plotted).



  2. Use smooth in combination of the appropriate sample points. The smooth key basically interpolates between two points using splines. Usually, a twice continuously differentiable spline (a cubic spline) is used and it is good enough to fool the human eyes. This is what I propose in my new answer.



  3. Use Gaussian or normal density approximation. There is a Local Central Limit Theorem which states




    The discrete binomial probabilities can be approximated by a continuous Gaussian/normal density curve: The larger the n*p and n*(1-p), the better the approximation. (cf. Probability: Theory and Examples (4th ed.) by Rick Durrett, Sections 3.1 and 3.5).




    This is what I propose in my old answer.



Please note that care must be taken when doing factorial calculations! If you pass real numbers into binom, then be prepared to get a “probability” bigger than one (cf. this question of mine). In your original graph, some “probabilities” seems to be bigger than two!



New answer



I just realized the OP wanted a “normalized binomial probability mass plot” instead of a “normal density approximation plot”. In this case, you can tell TikZ to draw only the points at which x*n are integers by specifying samples=n+1. Remember to use integers for factorial calculations and to normalize in the coordinates afterward.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
endaxis

endtikzpicture
enddocument


binomial



Notice the top of the curve is around 0.2. Indeed, binom(11,15,0.7) = 0.218623131...



Old answer



Here, let’s use “Gaussian/normal approximation”. The approximating Gaussian/normal density takes the form



1/sqrt(2 * pi * n * p * (1-p)) * exp( - n * (x - p)^2 / (2 * p * (1-p)) )


Of course, this approximation works better if n*p and n*(1-p) are larger. If you are illustrating a normal approximation, then this would be your choice:



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
 declare function=normaldensity(x,n,p)=exp(-n*(x-p)^2/(2*p*(1-p)))/sqrt(2*pi*n*p*(1-p));
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
 addplot[orange, domain=0:1, smooth]normaldensity(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


normal



Here, the cyan curve is drawn using my new answer, and the orange curve is drawn using normal approximation.






share|improve this answer






















  • What formula is that, or how is it derived from the normalized binomial that I gave?
    – Sentient
    Sep 5 at 22:34










  • @Sentient Answer updated. ;-) I assume your naming normd is referring to “normal density”, correct?
    – Ruixi Zhang
    Sep 5 at 22:39











  • oh sorry, I just used normd as a shorten way to say normalized, since I wanted the normalized binomial. That was also a very good explanation of how you obtained the approximation -- thanks! :)
    – Sentient
    Sep 5 at 22:51











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
9
down vote



accepted










I get a very different result from Ruixi if I use the well-known relation Gamma(n+1)=n! to provide a continuous version of factorial. However, the result seems to match your Desmos curve very accurately, so I think it is correct. The definition of the Gamma function is from this answer, and I used it to provide a continuous generalization of factorial here, when encountering the same problem.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[% gamma definition from https://tex.stackexchange.com/a/120449/121799
 declare function=gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[blue, domain=0:1,samples=100,smooth]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



CROSS CHECKS: I first check that the Gamma function has been implemented correctly and then draw your contour with it.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=1:10,samples=10,only marks,mark=*]x!;
 addplot[blue, domain=1:10]gamma(x+1);
endaxis
endtikzpicture


begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
 addplot[blue, domain=0:1]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



As you see, the smoothed out version of your plot is very different from the one with factorial in, but factorial is per se not defined for non-integers.






share|improve this answer


















  • 1




    Oh, amazingly it looks like this would reproduce the Desmos (whatever that is) curve.
    – marmot
    Sep 5 at 22:47










  • Oof, I probably would've never though about using the Gamma function to smooth it out. Thanks!
    – Sentient
    Sep 5 at 22:52










  • @Sentient This is a standard trick which (essentially) gave t'Hooft a Nobel prize. ;-) You need it in order to compute loop diagrams in 4-epsilon dimensions, where epsilon is a real number.
    – marmot
    Sep 5 at 22:57














up vote
9
down vote



accepted










I get a very different result from Ruixi if I use the well-known relation Gamma(n+1)=n! to provide a continuous version of factorial. However, the result seems to match your Desmos curve very accurately, so I think it is correct. The definition of the Gamma function is from this answer, and I used it to provide a continuous generalization of factorial here, when encountering the same problem.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[% gamma definition from https://tex.stackexchange.com/a/120449/121799
 declare function=gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[blue, domain=0:1,samples=100,smooth]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



CROSS CHECKS: I first check that the Gamma function has been implemented correctly and then draw your contour with it.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=1:10,samples=10,only marks,mark=*]x!;
 addplot[blue, domain=1:10]gamma(x+1);
endaxis
endtikzpicture


begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
 addplot[blue, domain=0:1]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



As you see, the smoothed out version of your plot is very different from the one with factorial in, but factorial is per se not defined for non-integers.






share|improve this answer


















  • 1




    Oh, amazingly it looks like this would reproduce the Desmos (whatever that is) curve.
    – marmot
    Sep 5 at 22:47










  • Oof, I probably would've never though about using the Gamma function to smooth it out. Thanks!
    – Sentient
    Sep 5 at 22:52










  • @Sentient This is a standard trick which (essentially) gave t'Hooft a Nobel prize. ;-) You need it in order to compute loop diagrams in 4-epsilon dimensions, where epsilon is a real number.
    – marmot
    Sep 5 at 22:57












up vote
9
down vote



accepted







up vote
9
down vote



accepted






I get a very different result from Ruixi if I use the well-known relation Gamma(n+1)=n! to provide a continuous version of factorial. However, the result seems to match your Desmos curve very accurately, so I think it is correct. The definition of the Gamma function is from this answer, and I used it to provide a continuous generalization of factorial here, when encountering the same problem.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[% gamma definition from https://tex.stackexchange.com/a/120449/121799
 declare function=gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[blue, domain=0:1,samples=100,smooth]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



CROSS CHECKS: I first check that the Gamma function has been implemented correctly and then draw your contour with it.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=1:10,samples=10,only marks,mark=*]x!;
 addplot[blue, domain=1:10]gamma(x+1);
endaxis
endtikzpicture


begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
 addplot[blue, domain=0:1]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



As you see, the smoothed out version of your plot is very different from the one with factorial in, but factorial is per se not defined for non-integers.






share|improve this answer














I get a very different result from Ruixi if I use the well-known relation Gamma(n+1)=n! to provide a continuous version of factorial. However, the result seems to match your Desmos curve very accurately, so I think it is correct. The definition of the Gamma function is from this answer, and I used it to provide a continuous generalization of factorial here, when encountering the same problem.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[% gamma definition from https://tex.stackexchange.com/a/120449/121799
 declare function=gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[blue, domain=0:1,samples=100,smooth]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



CROSS CHECKS: I first check that the Gamma function has been implemented correctly and then draw your contour with it.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.16

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=1:10,samples=10,only marks,mark=*]x!;
 addplot[blue, domain=1:10]gamma(x+1);
endaxis
endtikzpicture


begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);
 normd(x,n,p)=binom(x*n,n,p);
 gamma(z)=(2.506628274631*sqrt(1/z) + 0.20888568*(1/z)^(1.5) +
 0.00870357*(1/z)^(2.5) - (174.2106599*(1/z)^(3.5))/25920 -
 (715.6423511*(1/z)^(4.5))/1244160)*exp((-ln(1/z)-1)*z);
 smoothbinom(x,n,p)=gamma(n+1)/(gamma(x+1)*gamma((n-x)+1))*pow(p,x)*(1-p)^(n-x);
 smoothnormd(x,n,p)=smoothbinom(x*n,n,p);
 ]

beginaxis
 addplot[cyan, domain=0:1]normd(x, 15, 0.7);
 addplot[blue, domain=0:1]smoothnormd(x, 15, 0.7);
endaxis
endtikzpicture
enddocument


enter image description here



As you see, the smoothed out version of your plot is very different from the one with factorial in, but factorial is per se not defined for non-integers.







share|improve this answer














share|improve this answer



share|improve this answer








edited Sep 5 at 22:54

























answered Sep 5 at 22:46









marmot

56.5k462124




56.5k462124







  • 1




    Oh, amazingly it looks like this would reproduce the Desmos (whatever that is) curve.
    – marmot
    Sep 5 at 22:47










  • Oof, I probably would've never though about using the Gamma function to smooth it out. Thanks!
    – Sentient
    Sep 5 at 22:52










  • @Sentient This is a standard trick which (essentially) gave t'Hooft a Nobel prize. ;-) You need it in order to compute loop diagrams in 4-epsilon dimensions, where epsilon is a real number.
    – marmot
    Sep 5 at 22:57












  • 1




    Oh, amazingly it looks like this would reproduce the Desmos (whatever that is) curve.
    – marmot
    Sep 5 at 22:47










  • Oof, I probably would've never though about using the Gamma function to smooth it out. Thanks!
    – Sentient
    Sep 5 at 22:52










  • @Sentient This is a standard trick which (essentially) gave t'Hooft a Nobel prize. ;-) You need it in order to compute loop diagrams in 4-epsilon dimensions, where epsilon is a real number.
    – marmot
    Sep 5 at 22:57







1




1




Oh, amazingly it looks like this would reproduce the Desmos (whatever that is) curve.
– marmot
Sep 5 at 22:47




Oh, amazingly it looks like this would reproduce the Desmos (whatever that is) curve.
– marmot
Sep 5 at 22:47












Oof, I probably would've never though about using the Gamma function to smooth it out. Thanks!
– Sentient
Sep 5 at 22:52




Oof, I probably would've never though about using the Gamma function to smooth it out. Thanks!
– Sentient
Sep 5 at 22:52












@Sentient This is a standard trick which (essentially) gave t'Hooft a Nobel prize. ;-) You need it in order to compute loop diagrams in 4-epsilon dimensions, where epsilon is a real number.
– marmot
Sep 5 at 22:57




@Sentient This is a standard trick which (essentially) gave t'Hooft a Nobel prize. ;-) You need it in order to compute loop diagrams in 4-epsilon dimensions, where epsilon is a real number.
– marmot
Sep 5 at 22:57










up vote
6
down vote













Added: Please note that the following discussions are based on my background in probability theory. In your example, binomial probabilities are defined only when x*n are integers; that is, these probabilities are defined only when x is equal to one of the following 16 numbers



0, 1/15, 2/15, ... , 14/15, 15/15


Thus, in order to smoothly extend the graph, you have many options. These extensions include but are not limited to




  1. Analytical continuation via the gamma function. But, you should be aware of the fact that




    TeX is no Mathematica or MATLAB or Maple




    and therefore the gamma function is not available on the fly. So, @marmot provided an excellent answer in which the gamma function is hardcoded using its asymptotic expansions. Because it uses asymptotic expansions, there could be unexpected computation errors (but only if you choose extreme domain to be plotted).



  2. Use smooth in combination of the appropriate sample points. The smooth key basically interpolates between two points using splines. Usually, a twice continuously differentiable spline (a cubic spline) is used and it is good enough to fool the human eyes. This is what I propose in my new answer.



  3. Use Gaussian or normal density approximation. There is a Local Central Limit Theorem which states




    The discrete binomial probabilities can be approximated by a continuous Gaussian/normal density curve: The larger the n*p and n*(1-p), the better the approximation. (cf. Probability: Theory and Examples (4th ed.) by Rick Durrett, Sections 3.1 and 3.5).




    This is what I propose in my old answer.



Please note that care must be taken when doing factorial calculations! If you pass real numbers into binom, then be prepared to get a “probability” bigger than one (cf. this question of mine). In your original graph, some “probabilities” seems to be bigger than two!



New answer



I just realized the OP wanted a “normalized binomial probability mass plot” instead of a “normal density approximation plot”. In this case, you can tell TikZ to draw only the points at which x*n are integers by specifying samples=n+1. Remember to use integers for factorial calculations and to normalize in the coordinates afterward.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
endaxis

endtikzpicture
enddocument


binomial



Notice the top of the curve is around 0.2. Indeed, binom(11,15,0.7) = 0.218623131...



Old answer



Here, let’s use “Gaussian/normal approximation”. The approximating Gaussian/normal density takes the form



1/sqrt(2 * pi * n * p * (1-p)) * exp( - n * (x - p)^2 / (2 * p * (1-p)) )


Of course, this approximation works better if n*p and n*(1-p) are larger. If you are illustrating a normal approximation, then this would be your choice:



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
 declare function=normaldensity(x,n,p)=exp(-n*(x-p)^2/(2*p*(1-p)))/sqrt(2*pi*n*p*(1-p));
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
 addplot[orange, domain=0:1, smooth]normaldensity(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


normal



Here, the cyan curve is drawn using my new answer, and the orange curve is drawn using normal approximation.






share|improve this answer






















  • What formula is that, or how is it derived from the normalized binomial that I gave?
    – Sentient
    Sep 5 at 22:34










  • @Sentient Answer updated. ;-) I assume your naming normd is referring to “normal density”, correct?
    – Ruixi Zhang
    Sep 5 at 22:39











  • oh sorry, I just used normd as a shorten way to say normalized, since I wanted the normalized binomial. That was also a very good explanation of how you obtained the approximation -- thanks! :)
    – Sentient
    Sep 5 at 22:51















up vote
6
down vote













Added: Please note that the following discussions are based on my background in probability theory. In your example, binomial probabilities are defined only when x*n are integers; that is, these probabilities are defined only when x is equal to one of the following 16 numbers



0, 1/15, 2/15, ... , 14/15, 15/15


Thus, in order to smoothly extend the graph, you have many options. These extensions include but are not limited to




  1. Analytical continuation via the gamma function. But, you should be aware of the fact that




    TeX is no Mathematica or MATLAB or Maple




    and therefore the gamma function is not available on the fly. So, @marmot provided an excellent answer in which the gamma function is hardcoded using its asymptotic expansions. Because it uses asymptotic expansions, there could be unexpected computation errors (but only if you choose extreme domain to be plotted).



  2. Use smooth in combination of the appropriate sample points. The smooth key basically interpolates between two points using splines. Usually, a twice continuously differentiable spline (a cubic spline) is used and it is good enough to fool the human eyes. This is what I propose in my new answer.



  3. Use Gaussian or normal density approximation. There is a Local Central Limit Theorem which states




    The discrete binomial probabilities can be approximated by a continuous Gaussian/normal density curve: The larger the n*p and n*(1-p), the better the approximation. (cf. Probability: Theory and Examples (4th ed.) by Rick Durrett, Sections 3.1 and 3.5).




    This is what I propose in my old answer.



Please note that care must be taken when doing factorial calculations! If you pass real numbers into binom, then be prepared to get a “probability” bigger than one (cf. this question of mine). In your original graph, some “probabilities” seems to be bigger than two!



New answer



I just realized the OP wanted a “normalized binomial probability mass plot” instead of a “normal density approximation plot”. In this case, you can tell TikZ to draw only the points at which x*n are integers by specifying samples=n+1. Remember to use integers for factorial calculations and to normalize in the coordinates afterward.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
endaxis

endtikzpicture
enddocument


binomial



Notice the top of the curve is around 0.2. Indeed, binom(11,15,0.7) = 0.218623131...



Old answer



Here, let’s use “Gaussian/normal approximation”. The approximating Gaussian/normal density takes the form



1/sqrt(2 * pi * n * p * (1-p)) * exp( - n * (x - p)^2 / (2 * p * (1-p)) )


Of course, this approximation works better if n*p and n*(1-p) are larger. If you are illustrating a normal approximation, then this would be your choice:



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
 declare function=normaldensity(x,n,p)=exp(-n*(x-p)^2/(2*p*(1-p)))/sqrt(2*pi*n*p*(1-p));
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
 addplot[orange, domain=0:1, smooth]normaldensity(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


normal



Here, the cyan curve is drawn using my new answer, and the orange curve is drawn using normal approximation.






share|improve this answer






















  • What formula is that, or how is it derived from the normalized binomial that I gave?
    – Sentient
    Sep 5 at 22:34










  • @Sentient Answer updated. ;-) I assume your naming normd is referring to “normal density”, correct?
    – Ruixi Zhang
    Sep 5 at 22:39











  • oh sorry, I just used normd as a shorten way to say normalized, since I wanted the normalized binomial. That was also a very good explanation of how you obtained the approximation -- thanks! :)
    – Sentient
    Sep 5 at 22:51













up vote
6
down vote










up vote
6
down vote









Added: Please note that the following discussions are based on my background in probability theory. In your example, binomial probabilities are defined only when x*n are integers; that is, these probabilities are defined only when x is equal to one of the following 16 numbers



0, 1/15, 2/15, ... , 14/15, 15/15


Thus, in order to smoothly extend the graph, you have many options. These extensions include but are not limited to




  1. Analytical continuation via the gamma function. But, you should be aware of the fact that




    TeX is no Mathematica or MATLAB or Maple




    and therefore the gamma function is not available on the fly. So, @marmot provided an excellent answer in which the gamma function is hardcoded using its asymptotic expansions. Because it uses asymptotic expansions, there could be unexpected computation errors (but only if you choose extreme domain to be plotted).



  2. Use smooth in combination of the appropriate sample points. The smooth key basically interpolates between two points using splines. Usually, a twice continuously differentiable spline (a cubic spline) is used and it is good enough to fool the human eyes. This is what I propose in my new answer.



  3. Use Gaussian or normal density approximation. There is a Local Central Limit Theorem which states




    The discrete binomial probabilities can be approximated by a continuous Gaussian/normal density curve: The larger the n*p and n*(1-p), the better the approximation. (cf. Probability: Theory and Examples (4th ed.) by Rick Durrett, Sections 3.1 and 3.5).




    This is what I propose in my old answer.



Please note that care must be taken when doing factorial calculations! If you pass real numbers into binom, then be prepared to get a “probability” bigger than one (cf. this question of mine). In your original graph, some “probabilities” seems to be bigger than two!



New answer



I just realized the OP wanted a “normalized binomial probability mass plot” instead of a “normal density approximation plot”. In this case, you can tell TikZ to draw only the points at which x*n are integers by specifying samples=n+1. Remember to use integers for factorial calculations and to normalize in the coordinates afterward.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
endaxis

endtikzpicture
enddocument


binomial



Notice the top of the curve is around 0.2. Indeed, binom(11,15,0.7) = 0.218623131...



Old answer



Here, let’s use “Gaussian/normal approximation”. The approximating Gaussian/normal density takes the form



1/sqrt(2 * pi * n * p * (1-p)) * exp( - n * (x - p)^2 / (2 * p * (1-p)) )


Of course, this approximation works better if n*p and n*(1-p) are larger. If you are illustrating a normal approximation, then this would be your choice:



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
 declare function=normaldensity(x,n,p)=exp(-n*(x-p)^2/(2*p*(1-p)))/sqrt(2*pi*n*p*(1-p));
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
 addplot[orange, domain=0:1, smooth]normaldensity(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


normal



Here, the cyan curve is drawn using my new answer, and the orange curve is drawn using normal approximation.






share|improve this answer














Added: Please note that the following discussions are based on my background in probability theory. In your example, binomial probabilities are defined only when x*n are integers; that is, these probabilities are defined only when x is equal to one of the following 16 numbers



0, 1/15, 2/15, ... , 14/15, 15/15


Thus, in order to smoothly extend the graph, you have many options. These extensions include but are not limited to




  1. Analytical continuation via the gamma function. But, you should be aware of the fact that




    TeX is no Mathematica or MATLAB or Maple




    and therefore the gamma function is not available on the fly. So, @marmot provided an excellent answer in which the gamma function is hardcoded using its asymptotic expansions. Because it uses asymptotic expansions, there could be unexpected computation errors (but only if you choose extreme domain to be plotted).



  2. Use smooth in combination of the appropriate sample points. The smooth key basically interpolates between two points using splines. Usually, a twice continuously differentiable spline (a cubic spline) is used and it is good enough to fool the human eyes. This is what I propose in my new answer.



  3. Use Gaussian or normal density approximation. There is a Local Central Limit Theorem which states




    The discrete binomial probabilities can be approximated by a continuous Gaussian/normal density curve: The larger the n*p and n*(1-p), the better the approximation. (cf. Probability: Theory and Examples (4th ed.) by Rick Durrett, Sections 3.1 and 3.5).




    This is what I propose in my old answer.



Please note that care must be taken when doing factorial calculations! If you pass real numbers into binom, then be prepared to get a “probability” bigger than one (cf. this question of mine). In your original graph, some “probabilities” seems to be bigger than two!



New answer



I just realized the OP wanted a “normalized binomial probability mass plot” instead of a “normal density approximation plot”. In this case, you can tell TikZ to draw only the points at which x*n are integers by specifying samples=n+1. Remember to use integers for factorial calculations and to normalize in the coordinates afterward.



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
endaxis

endtikzpicture
enddocument


binomial



Notice the top of the curve is around 0.2. Indeed, binom(11,15,0.7) = 0.218623131...



Old answer



Here, let’s use “Gaussian/normal approximation”. The approximating Gaussian/normal density takes the form



1/sqrt(2 * pi * n * p * (1-p)) * exp( - n * (x - p)^2 / (2 * p * (1-p)) )


Of course, this approximation works better if n*p and n*(1-p) are larger. If you are illustrating a normal approximation, then this would be your choice:



documentclassarticle
usepackagepgfplots
pgfplotssetcompat=1.7

% https://tex.stackexchange.com/questions/198572/tikz-binomial-distribution

begindocument
begintikzpicture[
 declare function=binom(x,n,p)=n!/(x!*(n-x)!)*p^x*(1-p)^(n-x);,
% declare function=normd(x,n,p)=binom(x*n,n,p);% <- This is bad practice
 declare function=normaldensity(x,n,p)=exp(-n*(x-p)^2/(2*p*(1-p)))/sqrt(2*pi*n*p*(1-p));
]

beginaxis
 addplot[cyan, samples at=0,1,...,15, smooth](x/15,binom(x, 15, 0.7));
 addplot[orange, domain=0:1, smooth]normaldensity(x, 15, 0.7);
endaxis

endtikzpicture
enddocument


normal



Here, the cyan curve is drawn using my new answer, and the orange curve is drawn using normal approximation.







share|improve this answer














share|improve this answer



share|improve this answer








edited Sep 6 at 4:57

























answered Sep 5 at 22:30









Ruixi Zhang

3,271216




3,271216











  • What formula is that, or how is it derived from the normalized binomial that I gave?
    – Sentient
    Sep 5 at 22:34










  • @Sentient Answer updated. ;-) I assume your naming normd is referring to “normal density”, correct?
    – Ruixi Zhang
    Sep 5 at 22:39











  • oh sorry, I just used normd as a shorten way to say normalized, since I wanted the normalized binomial. That was also a very good explanation of how you obtained the approximation -- thanks! :)
    – Sentient
    Sep 5 at 22:51

















  • What formula is that, or how is it derived from the normalized binomial that I gave?
    – Sentient
    Sep 5 at 22:34










  • @Sentient Answer updated. ;-) I assume your naming normd is referring to “normal density”, correct?
    – Ruixi Zhang
    Sep 5 at 22:39











  • oh sorry, I just used normd as a shorten way to say normalized, since I wanted the normalized binomial. That was also a very good explanation of how you obtained the approximation -- thanks! :)
    – Sentient
    Sep 5 at 22:51
















What formula is that, or how is it derived from the normalized binomial that I gave?
– Sentient
Sep 5 at 22:34




What formula is that, or how is it derived from the normalized binomial that I gave?
– Sentient
Sep 5 at 22:34












@Sentient Answer updated. ;-) I assume your naming normd is referring to “normal density”, correct?
– Ruixi Zhang
Sep 5 at 22:39





@Sentient Answer updated. ;-) I assume your naming normd is referring to “normal density”, correct?
– Ruixi Zhang
Sep 5 at 22:39













oh sorry, I just used normd as a shorten way to say normalized, since I wanted the normalized binomial. That was also a very good explanation of how you obtained the approximation -- thanks! :)
– Sentient
Sep 5 at 22:51





oh sorry, I just used normd as a shorten way to say normalized, since I wanted the normalized binomial. That was also a very good explanation of how you obtained the approximation -- thanks! :)
– Sentient
Sep 5 at 22:51


















 

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