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I have been given a binomial distribution: $$B(m+n;n,p)=\frac{(m+n)!}{m!n!}p^mq^n.$$

Here $m = 10^3$, $n=10^2$, $p=10^{-2}$, $q=1-p.$

I'm using MATLAB to compute log $B(m+n;n,p)$ and store the value in logB

m=10^3;
n=10^2;
p=10^(-2);
q=1-p;
logB=log(factorial(m+n)/(factorial(m)*factorial(n))*p^m*q^n)

I get logB as NaN . How can I modify the formula to avoid floating point error and get a valid answer?

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    $\begingroup$ Taking those factorials is probably giving you infinities, the division of which gives you NaN. You need a factorial division function which preemptively cancels. $\endgroup$ – Richard Jun 2 at 13:53
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    $\begingroup$ You have a ratio of a product of $n$ terms by a product of $n$ terms. Write this as the product of $n$ ratios, each of which should of moderate size. $\endgroup$ – Wolfgang Bangerth Jun 2 at 14:24
  • $\begingroup$ @Richard I'm trying to simplify but still stuck: ((m+n)(m+n-1)...(m+1))/n! $\endgroup$ – Rhombus Jun 2 at 14:27
  • $\begingroup$ @WolfgangBangerth could you elaborate a bit? Thanks. $\endgroup$ – Rhombus Jun 2 at 14:43
  • $\begingroup$ You want to compute $\frac{abc}{def}$ where both enumerator and denominator end up very large numbers. So compute it as $\frac ad\cdot \frac be \cdot \frac cf$ where now each fraction is of moderate size. $\endgroup$ – Wolfgang Bangerth Jun 2 at 18:31
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This is why gammaln exists

logB = gammaln(m+n+1) - gammaln(m+1) - gammaln(n+1) + m*log(p) + n*log(q);

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