Bernoulli and Binomial

Bernoulli Trials & Random Variables

Definition:[Bernoulli Trails]
A Bernoulli trial is a simple random variable with only two outcomes, one outcome is success denoted by $s$ , and the other is failure, denoted by $f$ .

Definition[Bernoulli Random Variable]
The sample space of a Bernoulli trial consists of $s$ and $f$ , the random variable defined by $X(s) = 1$ and $X(f) = 0$ is called a \textit{Bernoulli random variable}. If we denote the probability of success by $p$ the probability mass function of $X$ is

$\begin{align*} p(x) = \begin{cases} 1 - p = q & \text{if } x = 0 \\ p & \text{if } x = 1 \\ 0 & \text{otherwise} \end{cases} \end{align*}$

A random variable is called Bernoulli with parameter $p$ if it’s probability mass function is given by above. The expected value, variance, and standard deviation of a Bernoulli random variable is given by

$\begin{align*} E[X] &= p \\ V[X] &= p(1 - p) \\ \sigma_X &= \sqrt{p(1-p)} \end{align*}$

Example:
If in a throw of a fair dice the event of a 4 or 6 is consider a success, describe the pmf for the related Bernoulli random variable.

Example:
In a county hospital 10 babies, of whome six were boys were born last Thursday. What is the probability that the first six births were all boys?

Binomial Random Variables

Definition[Binomial Random Variable]
If $n$ Bernoulli trails all with probability of success $p$ are performed \textbf{independently}, then $X$ , the number of successes is called \textbf{binomial with parameters $n$ and $p$ }. The set of possible values of $X$ is $\set{0,1,2,...,n}$ , and has the probability mass function given by

$\begin{align*} b(x;n,p) = p(x) = P(X = x) = \begin{cases} \dbinom{n}{x}p^x(1-p)^{n-x} & \text{if } x = 0,1,2,...,n \\ 0 &\text{otherwise} \end{cases} \end{align*}$

The expectations, variances, and standard deviations of a binomial random variable with parameters $n$ and $p$ are

$\begin{align*} E[X] &= np \\ V[X] &= np(1-p) \\ \sigma_X &= \sqrt{np(1-p)} \end{align*}$