Binomial distribution

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The binomial distribution $B(n,p)$ represents the probability of success on $n$ trials with a probability of $p$.

  • The mass function is
\[ \begin{split}\begin{cases} \mathbb{P}(X=k) = 0 & if & k > n \\ \mathbb{P}(X=k) = C_n^k * p^k * (1-p)^{n-k} & else \\ \end{cases}\end{split} \]
  • $\mathbb{E}(X) = \ n * p$
  • $\mathbb{V}(X) = \ n * p * (1-p)$

The probability of having $k$ successes on $n$ trials means that

  • we got $k$ successes
  • we got $n-k$ failures (the remaining trials)

So we have the probability

  • $p^k$ because we want $k$ successes with p the probability of success
  • $(1-p)^{(n-k)}$ because we want $n-k$ failures and $1-p$ if the probability of failure.
  • and since we don't care about the order, we need to multiply by the permutations $C_n^k$