¶ Bernoulli Experiment

A Bernoulli experiment, more commonly called a Bernoulli trial, is a random experiment, the outcome of which can be classified in but one of two mutually exclusive and exhaustive ways, for instance, success or failure (e.g., female or male, life or death, nondefective or defective).

A sequence of independent Bernoulli trials occurs when a Bernoulli experiment is performed several independent times and the probability of success, say $pp$, remains the same from trial to trial.

¶ Example

In the Michigan daily lottery the probability of winning when placing a six-way boxed bet is $0.0060.006$. Assuming independence, a bet placed on each of $1212$ successive days would correspond to $1212$ independent Bernoulli trials with $p=0.006p = 0.006$.

¶ Bernoulli Distribution

Let $XX$ be a random variable associated with a Bernoulli trial by defining it as follows:

That is, the two outcomes, success and failure, are denoted by one and zero, respectively. The pmf of $XX$ can be written as

and we say that $XX$ has a Bernoulli Distribution.

In a sequence of $nn$ Bernoulli trials, we shall let ${X}_{i}X_i$ denote the Bernoulli random variable associated with the $ii$th trial. An observed sequence of $nn$ independent Bernoulli trials will then be an $nn$-tuple of zeros and ones. We often call this collection a random sample of size $nn$ from a Bernoulli distribution.

¶ Binomial Distribution

In a sequence of Bernoulli trials, we are often interested in the total number of successes but not the actual order of their occurrences. If we let the random variable $XX$ equal the number of observed successes in $nn$ Bernoulli trials, then the possible values of $XX$ are $0,1,2,\cdots \text{\hspace{0.17em}},n0, 1, 2, \cdots , n$. If $xx$ successes occur, where $x=0,1,2,\cdots \text{\hspace{0.17em}},nx = 0, 1, 2, \cdots , n$, then $n\text{−}xn - x$ failures occur. The number of ways of selecting $xx$ positions for the $xx$ successes in the $nn$ trials is

$\left(\genfrac{}{}{0px}{}{n}{x}\right)=\frac{n!}{x!\left(n-x\right)!}\binom\left\{n\right\}\left\{x\right\}=\frac\left\{n!\right\}\left\{x!\left(n-x\right)!\right\}$

Since the trials are independent and since the probabilities of success and failure on each trial are, respectively, $pp$ and $q=1\text{−}pq = 1 - p$, the probability of each of these ways is ${p}^{x}\left(1\text{−}p{\right)}^{n\text{−}x}p^x \left(1 - p\right)^\left\{n-x\right\}$. Thus, $f\left(x\right)f\left(x\right)$, the pmf of $XX$, is the sum of the probabilities of the $\left(\genfrac{}{}{0px}{}{n}{x}\right)\displaystyle\binom\left\{n\right\}\left\{x\right\}$ mutually exclusive events; that is,

These probabilities are called binomial probabilities, and the random variable X is said to have a binomial distribution.

Summarizing, a binomial experiment satisfies the following properties:

• A Bernoulli (success–failure) experiment is performed $nn$ times, where $nn$ is a (non-random) constant.
• The trials are independent.
• The probability of success on each trial is a constant $pp$; the probability of failure is $q=1\text{−}pq = 1- p$.
• The random variable $XX$ equals the number of successes in the $nn$ trials.

A binomial distribution will be denoted by the symbol $b\left(n,p\right)b\left(n, p\right)$, and we say that the distribution of $XX$ is $b\left(n,p\right)b\left(n, p\right)$. The constants $nn$ and $pp$ are called the parameters of the binomial distribution; they correspond to the number $nn$ of independent trials and the probability $pp$ of success on each trial.

The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so $n=1n=1$ for such a binomial distribution).