Discrete Probability Distributions - Binomial Distribution

Discrete Probability Distributions

Binomial Random Variable

A random variable is binomial if it meets the following criteria:

• It measures the number of successes in $n$ identical trials.
• Each trial has only two possible outcomes: success or failure.
• The probability of success (and failure) remains constant from one trial to another.
• Trials are independent.

Example 1: Pat's Quiz

Pat, a statistics student, relies on guessing for a quiz with 10 multiple-choice questions, each having 5 possible answers with only 1 correct.

We want to find the probability that:

• Pat answers all questions incorrectly.
• Pat answers exactly 2 questions correctly.
• Pat fails the quiz (score less than 50%, each question worth 1 point).

Example 2: Non-Binomial Scenario

Pat attempts to solve each question rather than guess. However, solving statistics questions is very brain-consuming and Pat gets tired, this means that the chance of getting each question right decreases as he works on the questions. This will not be a binomial experiment because the probability of success does not remain constant.

Bernoulli Distribution

Let $B$ be a random variable that takes the value 1 with probability $\pi$ and the value 0 with probability $1 - \pi$.

• Mean of $B$:
  $\mu_B = E[B] = \sum bP(b) = 0 \times (1 - \pi) + 1 \times \pi = \pi$
• Variance of $B$:
  $\text{Var}(B) = \sum (b - \mu_B)^2 p(b) = (0 - \pi)^2 (1 - \pi) + (1 - \pi)^2 \pi = \pi^2 (1 - \pi) + (1 - 2\pi + \pi^2) \pi = \pi^2 - \pi^3 + \pi - 2\pi^2 + \pi^3 = \pi - \pi^2 = \pi(1 - \pi)$

Binomial Distribution

Let $X$ be a random variable that follows the binomial distribution with parameters $n$ and $\pi$, denoted as $X \sim Bin(n, \pi)$.

• $X$ is the sum of $n$ Bernoulli random variables with parameter $\pi$:
  $X = B<em>1 + B</em>2 + \cdots + B_n$
• Mean of $X$:
  $\mu<em>X = E[X] = E[B</em>1] + E[B<em>2] + \cdots + E[B</em>n] = n\mu_B = n\pi$
• Variance of $X$:
  $\text{Var}(X) = \text{Var}(B<em>1 + B</em>2 + \cdots + B<em>n)$, where $B</em>1, B<em>2, \text{etc.}$ are independent of one another, so we can apply the rule about the variance of the sum of independent random variables: $\text{Var}(X) = \text{Var}(B</em>1) + \text{Var}(B<em>2) + \cdots + \text{Var}(B</em>n) = n\pi(1 - \pi)$

Example: Number of Boys in a 6-Children Family

Let $X$ = number of boys in families with 6 children where $n = 6$ and $\pi = 0.5$ (assuming each birth has a 50% chance of being a boy).

Probability Distribution

Mean Calculation

$\mu = \sum x p(x) = n \pi$

In this example: $\mu = 6 \times 0.5 = 3$

Variance Calculation

$\sigma^2 = \sum (x - \mu)^2 p(x) = n \pi (1 - \pi)$

In this example: $\sigma^2 = 6 \times 0.5 \times (1 - 0.5) = 1.5$

Pat's Quiz - Population Mean and Standard Deviation

Suppose that a professor has a class of Pat. What is the population mean? What is the population standard deviation?

Binomial Distribution Table

Table 5 in Appendix B of the textbook provides binomial distribution probabilities.

Essentials of Statistics For Business and Economics, Anderson et al. 9th Edition, pages 813 to 821.