binomial distribution — Jemoka Knowledge Base

A binomial distribution is a typo of distribution whose contents are: Binary Independent Fixed number Same probability: “That means: WITH REPLACEMENT” Think: “what’s the probability of n coin flips getting k heads given the head’s probability is p”. constituents We write:

\begin{equation} X \sim Bin(n,p) \end{equation}

where, n is the number of trials, p is the probability of success on each trial. requirements Here is the probability mass function:

\begin{equation} P(X=k) = {n \choose k} p^{k}(1-p)^{n-k} \end{equation}

additional information properties of binomial distribution expected value: np variance: np(1-p) deriving the expectation The expectation of the binomial distribution is derivable from the fact:

\begin{equation} X = \sum_{i=1}^{n} Y_{i} \end{equation}

where,

\begin{equation} \begin{cases} X \sim Bin(n,p) \\ Y_{i} \sim Bern(p) \end{cases} \end{equation}

Now, recall that expected value is linear. Therefore, we can write that: approximating binomial normal distribution approximation: n > 20, variance large (np(1-p)) > 10, absolute independence; beware of continuity correction poisson distribution approximation: n > 20, p small p < 0.05 adding binomial distribution For X and Y independent binomial distributions, with equivalent probability:

\begin{equation} X \sim Bin(a, p), Y \sim Bin(b, p) \end{equation}

Then:

\begin{equation} X+Y \sim Bin(a+b, p) \end{equation}