Chapter 5: Probability Distributions – Binomial and Poisson Distributions

Probability Distributions

Introduction

Probability distributions describe how probabilities are distributed over the values of a random variable. In statistics, discrete probability distributions such as the binomial and Poisson distributions are fundamental for modeling random processes and interpreting statistical results.

Binomial Probability Distributions

Definition and Requirements

• Binomial Probability Distribution: A discrete probability distribution resulting from a procedure that meets these four requirements:

  1. The procedure has a fixed number of trials (n).

  2. The trials are independent.

  3. Each trial has exactly two categories (success and failure).

  4. The probability of success (p) remains the same in all trials.

Notation

• S and F: Success and failure, the two possible outcomes.

• P(S) = p: Probability of success in one trial.

• P(F) = 1 - p = q: Probability of failure in one trial.

• n: Fixed number of trials.

• x: Number of successes in n trials (0 ≤ x ≤ n).

• P(x): Probability of getting exactly x successes in n trials.

Example: Cashless Adults

• Suppose the probability that a randomly selected adult smartphone owner is cashless is 0.05.

• We want the probability that, among 10 randomly selected adults, exactly 2 are cashless.

• This scenario meets all binomial requirements: fixed number of trials (n = 10), independence, two outcomes (cashless or not), and constant probability (p = 0.05).

• Values: n = 10, x = 2, p = 0.05, q = 0.95.

Binomial Probability Formula

The probability of exactly x successes in n trials is given by:

$P(x) = \frac{n!}{(n-x)!x!} \cdot p^x \cdot q^{n-x}$

• Where n = number of trials, x = number of successes, p = probability of success, q = probability of failure.

Example Calculation

Given n = 10, x = 2, p = 0.05, q = 0.95:

$P(2) = \frac{10!}{(10-2)!2!} \cdot 0.05^2 \cdot 0.95^{8}$ $= \frac{10!}{8!2!} \cdot 0.0025 \cdot 0.663420$ $= 45 \cdot 0.0025 \cdot 0.663420 = 0.074635$ $= 0.0746$ (rounded to three significant digits)

The probability of getting exactly two cashless adults is 0.0746.

Alternative Methods

• Excel: The BINOM.DIST function can be used to compute binomial probabilities efficiently.

Mean and Standard Deviation of Binomial Distributions

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

• Standard Deviation: $\sigma = \sqrt{npq}$

Range Rule of Thumb

• Significantly low values: $\leq (\mu - 2\sigma)$

• Significantly high values: $\geq (\mu + 2\sigma)$

• Values not significant: Between $(\mu - 2\sigma)$ and $(\mu + 2\sigma)$

Example: NFL Overtime Wins

• n = 460 games, p = 0.5, q = 0.5

• Mean: $\mu = 460 \times 0.5 = 230.0$ games

• Standard deviation: $\sigma = \sqrt{460 \times 0.5 \times 0.5} = 10.7$ games

• Significantly low: $230.0 - 2(10.7) = 208.6$ games

• Significantly high: $230.0 + 2(10.7) = 251.4$ games

• 252 wins is significantly high (greater than 251.4).

Poisson Probability Distributions

Definition and Requirements

• Poisson Probability Distribution: A discrete probability distribution that applies to the number of occurrences of an event over a specified interval (time, distance, area, volume, etc.).

• Requirements:

  1. The random variable x is the number of occurrences in some interval.

  2. The occurrences must be random.

  3. The occurrences must be independent.

  4. The occurrences must be uniformly distributed over the interval.

Poisson Probability Formula

The probability of the event occurring x times over an interval is:

$P(x) = \frac{\mu^x \cdot e^{-\mu}}{x!}$

• Where $e \approx 2.71828$, $\mu$ = mean number of occurrences in the interval.

Parameters and Properties

• Mean: $\mu$

• Standard deviation: $\sigma = \sqrt{\mu}$

• A Poisson distribution is determined only by its mean $\mu$.

• Possible x values: 0, 1, 2, ... (no upper limit).

Example: Atlantic Hurricanes

• 652 hurricanes in 118 years: $\mu = 652/118 = 5.5$ hurricanes/year.

• Probability of exactly 6 hurricanes in a year:

$P(6) = \frac{5.5^6 \cdot 2.71828^{-5.5}}{6!}$ $= \frac{27,680.64 \cdot 0.0040867866}{720} = 0.157$

• Expected number of years with 6 hurricanes: $118 \times 0.157 = 18.5$ years (actual: 16 years). The Poisson model fits well.

Poisson as an Approximation to Binomial

• The Poisson distribution can approximate the binomial distribution when:

  1. n is large ($n \geq 100$)

  2. p is small ($np \leq 10$)

• For the approximation, use $\mu = np$ in the Poisson formula.

Summary Table: Binomial vs. Poisson Distributions

Feature	Binomial Distribution	Poisson Distribution
Type	Discrete	Discrete
Number of Trials	Fixed (n)	Not fixed (interval-based)
Probability of Success	Constant (p)	Small, events rare
Parameter(s)	n, p	$\mu$ (mean)
Formula	$P(x) = \frac{n!}{(n-x)!x!} p^x q^{n-x}$	$P(x) = \frac{\mu^x e^{-\mu}}{x!}$

Additional info: The notes also reference the use of Excel for calculating binomial probabilities and emphasize the importance of interpreting probability values in context, such as determining whether observed results are significantly high or low using the range rule of thumb.