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Binomial Distribution

docs/tutorial:stats:discrete_binom

A binomial random variable with parameters $(n, p)$ can be described as the sum of $n$ independent Bernoulli random variables of parameter $p;$

Y = i = 1 \sum n X_{i} .

Therefore, this random variable counts the number of successes in $n$ independent trials of a random experiment where the probability of success is $p .$

where the incomplete beta integral is

I_{x} (a, b) = \frac{Γ ( a + b )}{Γ ( a ) Γ ( b )} \int_{0}^{x} t^{a - 1} (1 - t)^{b - 1} d t .

Now

M (t) = [1 - p (1 - e^{t})]^{n}

Implementation: scipy.stats.binom