Bernoulli and Binomial Distributions

Bernoulli Distribution

An r.v. $X$ is said to have the Bernoulli distribution with parameter $p$ if $P(X=1)=p$ and $P(X=0)=1-p$, where $0<p<1$. We write this as $X\sim \mathrm{Bern}(p)$. (or $B(1,p)$, $\mathrm{Bin}(1,p)$ as a special case of binomial distribution)

Bernoulli Trial

An experiment that can result in either a "success" of a "failure" (but not both) is called Bernoulli trial. A Bernoulli random variable can be thought of as the indicator of success in a Bernoulli trial: it equals $1$ if success occurs and $0$ if failure occurs in the trail.

Binomial Distribution

Suppose that $n$ independent Bernoulli trials are performed, each with the same success probability $p$. Let $X$ be the number of successes. The distribution of $X$ is called the Binomial distribution with parameters $n$ and $p$. We write $X\sim \mathrm{Bin}(n,p)$ or $X\sim B(n,p)$ to mean that $X$ has the Binomial distribution with parameters $n$ is $p$, where $n$ is a positive integer and $0<p<1$

Properties

If $X\sim \mathrm{Bin}(n,p)$, then the PMF of $X$ is

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

for $k=0,1,\dots ,n$, with $\mathrm{E}[X]=np$ and $\mathrm{Var}(X)=np(1-p)$

Let $X\sim \mathrm{Bin}(n,p)$, and $q=1-p$, then $n-X\sim \mathrm{Bin}(n,q)$

Let $X\sim \mathrm{Bin}(n,p)$ with $p=\frac{1}{2}$ and $n$ even. Then the distribution of $X$ is symmetric about $\frac{n}{2}$, that is, $P(X=n/2+j)=P(X=n/2-j)$ for all nonnegative integers $j$

Regenerative Property

Let $X_1\sim \mathrm{Bin}(n,p)$ and $X_2\sim \mathrm{Bin}(m,p)$, if $X_1$ and $X_2$ are independent, we have $X_1+X_2\sim \mathrm{Bin}(n+m,p)$.

To generalize, if $X_i\sim \mathrm{Bin}(n_i,p)$ and $X_1,\dots ,X_k$ are independent, we can conclude

$$\sum _{i=1}^k{X}_i\sim \mathrm{Bin}\left(\sum _{i=1}^k{n}_i,p\right)$$

Negative Binomial Distribution (Pascal Distribution)

This distribution models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted $r$) occurs.

$$X\sim \mathrm{NB}(r,p)$$

The PMF of the negative binomial distribution is

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{k+r-1}{k}\right)(1-p{)}^k{p}^r$$

and $\mathrm{E}[X]=\frac{r}{p}-1$