Discrete Probability Distributions: Random Variables, Binomial, Geometric, and Poisson Distributions

Section 4.1 - Probability Distributions

Random Variables

A random variable is a numerical value assigned to each outcome of a probability experiment. Random variables are fundamental in probability and statistics, as they allow us to quantify outcomes and analyze their distributions.

• Notation: Random variables are typically denoted by x.

Types of Random Variables

• Discrete Random Variable: Has a finite or countable number of possible outcomes that can be listed (e.g., 0, 1, 2, 3, ...).

• Continuous Random Variable: Has an uncountable number of possible outcomes, represented by an interval on the number line (e.g., any value between 0 and 21 gallons).

Example:

• If x is the number of Fortune 500 companies that lost money in the previous year, x is discrete because it can be counted.

• If x is the volume of gasoline in a 21-gallon tank, x is continuous because it can take any value between 0 and 21 gallons.

Discrete Probability Distribution

A discrete probability distribution lists each possible value a discrete random variable can assume, together with its probability. It must satisfy two conditions:

• The probability of each value is between 0 and 1, inclusive: $0 \leq P(x) \leq 1$

• The sum of all probabilities is 1: $\sum P(x) = 1$

Constructing a Discrete Probability Distribution

1. List all possible outcomes of the discrete random variable.

2. Make a frequency distribution for these outcomes.

3. Find the sum of the frequencies.

4. Calculate the probability of each outcome by dividing its frequency by the total frequency.

5. Check that each probability is between 0 and 1 and that their sum is 1.

Mean, Variance, and Standard Deviation of a Discrete Probability Distribution

• Mean (Expected Value): $\mu = \sum x P(x)$

• Variance: $\sigma^2 = \sum (x - \mu)^2 P(x)$

• Standard Deviation: $\sigma = \sqrt{\sum (x - \mu)^2 P(x)}$

Expected Value: The expected value of a discrete random variable is equal to its mean: $E(x) = \mu = \sum x P(x)$

Section 4.2 - Binomial Distributions

Binomial Experiments

A binomial experiment is a probability experiment that satisfies the following conditions:

• The experiment is repeated for a fixed number of trials (n), and each trial is independent.

• Each trial has only two possible outcomes: success (S) or failure (F).

• The probability of success (p) is the same for each trial.

• The random variable x counts the number of successful trials.

Notation for Binomial Experiments

• n: Number of trials

• p: Probability of success in a single trial

• q: Probability of failure in a single trial ($q = 1 - p$)

• x: Number of successes in n trials ($x = 0, 1, 2, ..., n$)

Binomial Probability Formula

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

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

• n: Number of trials

• p: Probability of success

• q: Probability of failure ($q = 1 - p$)

• x: Number of successes

Example:

A survey found that 22% of U.S. adults say that the economy, unemployment, and jobs are the most important problems facing the U.S. today. You randomly select four adults and ask them whether the economy, unemployment, and jobs are the most important problems facing the U.S. today. Find the probability that:

1. Exactly two of them respond yes: Given $n = 4$, $p = 0.22$, $q = 0.78$, $x = 2$: $P(2) = {4 \choose 2} (0.22)^2 (0.78)^2 = 6 \times (0.22)^2 \times (0.78)^2 = 0.177$

2. At least two of them respond yes: Sum $P(2) + P(3) + P(4)$ (using the binomial formula for each value of x) Probability = 0.212

3. Fewer than two of them respond yes: Sum $P(0) + P(1)$ Probability = (calculated as above)

Mean, Variance, and Standard Deviation of a Binomial Distribution

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

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

Section 4.3 - More Discrete Probability Distributions

Geometric Distribution

The geometric distribution is a discrete probability distribution that models the number of trials needed to get the first success in repeated, independent Bernoulli trials (trials with two possible outcomes).

• Each trial is independent.

• The probability of success (p) is constant for each trial.

• The random variable x represents the trial on which the first success occurs.

• The probability that the first success occurs on trial x is: $P(x) = p q^{x-1}$, where $q = 1 - p$

Example: If the failure rate of businesses after five years is 50% ($p = 0.50$), and four businesses are selected at random, the probability that the fourth business selected is the first to have failed is: $P(4) = 0.50 \times (0.50)^{4-1} = 0.50 \times (0.50)^3 = 0.063$

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time, area, or volume, when these events happen with a known constant mean rate and independently of the time since the last event.

• The experiment consists of counting the number of times an event occurs in a given interval.

• The probability of the event occurring is the same for each interval.

• The number of occurrences in one interval is independent of the number in other intervals.

• The probability of exactly x occurrences in an interval is: $P(x) = \frac{\mu^x e^{-\mu}}{x!}$ where $e \approx 2.71818$ and $\mu$ is the mean number of occurrences.

Example: If the mean number of accidents per month at an intersection is three ($\mu = 3$), the probability that in any given month four accidents will occur is: $P(4) = \frac{3^4 (2.71818)^{-3}}{4!} = 0.168$

Additional info: The notes above expand on the original points by providing full definitions, formulas in LaTeX, and examples for each distribution. The standard deviation formula for the binomial distribution is corrected to $\sigma = \sqrt{npq}$, and for the general discrete distribution, $\sigma = \sqrt{\sum (x - \mu)^2 P(x)}$.