The Binomial Random Variable and Distribution

binomial random variable

X

Given a Binomial Experiment consisting of $n$ trials, the binomial random variable $X$ associated with this experiment is defined as:

X = the number of successes among the n trials

Suppose for example $n = 3$ . Then there are 8 possible outcomes of the experiment:

S = {S S S, S S F, S F S, S F F, F S S, F S F, F F S, F F F}

X \sim Bin (n, p)

We write $X \sim Bin (n, p)$ to indicate that $X$ is a binomial Random Variables based on $n$ trials with success probability $p$ . because the Probability Distribution, PMF of a binomial rv $X$ depends on the two parameters $n, p$ , we denote the binomial pmf by $b (x; n, p)$ .

Binomial rv probability

$b (x; n, p) = (\binom{n}{x}) p^{x} (1 - p)^{n - x} : x = 0, 1, 2, . . ., n$

Computing Binomial Probabilities

Notation

For $X \sim Bin (n, p)$ the cdf will be denoted by:

B (x; n, p) = P (X \leq x) = \sum_{y = 0}^{x} b (y; n, p) : x = 0, 1, 2, . . ., n

The idea is simple, that just:

P (X \leq x) = P (X = 0) + \dots + P (X = x)

So we just replace each $P$ with the appropriate $b (y; n, p)$ expansion.

The Mean, Variance, and Moment Generating Function

Proposition

If $X \sim Bin (n, p)$ then $E (X) = n p$ , $Var (X) = n p (1 - p)$ and $σ$ is just $\sqrt{n p (1 - p)}$ .

#todo/school : Finish these notes and the practice problems 📅 2024-10-09 ✅ 2024-10-09