Discrete Random Variables and Probability Distributions

Random Variables and Probability Distributions

Introduction to Random Variables

A random variable is a function that assigns a numerical value to each outcome of a random experiment. Random variables are fundamental in probability and statistics, as they allow us to quantify and analyze random phenomena.

• Discrete random variables can take on a countable number of distinct values (e.g., 0, 1, 2, ...).

• Continuous random variables can take on any value within a given interval.

Example: Tossing 3 coins. Let X = number of tails. Possible values for X are 0, 1, 2, or 3.

Probability Distribution of a Discrete Random Variable

The probability distribution of a discrete random variable X lists all possible values of X and the probability associated with each value. The sum of all probabilities must equal 1.

x	Outcomes	P(X = x)
0	HHH	1/8
1	HTH, THH, HHT	3/8
2	TTH, THT, HTT	3/8
3	TTT	1/8

Key Properties:

• Each probability P(X = x) must be between 0 and 1.

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

Examples of Discrete Random Variables

• Number of defective items in a production line (X = 0, 1, 2, ...)

• Number of forms with errors in a box of 20 tax returns (X = 0, 1, ..., 20)

• Number of sales from 10 cold calls (X = 0, 1, ..., 10)

Notation and Expected Value

The probability that X takes the value x is denoted as P(X = x).

The expected value (mean) of X, denoted E(X), is calculated as:

$E(X) = \sum_x x \; P(X = x)$

Example: In a fire insurance scenario, if the payout X is either 0 (no fire) or 200,000 (fire), and the probability of fire is 1/10,000, then:

$E(X) = 200,000 \times \frac{1}{10,000} + 0 \times \frac{9,999}{10,000} = 20$

Insurance companies set premiums higher than the expected payout to ensure profitability.

Variance and Standard Deviation

The variance of X, Var(X), measures the spread of the distribution:

$Var(X) = E(X^2) - [E(X)]^2$

The standard deviation of X, SD(X), is the square root of the variance:

$SD(X) = \sqrt{Var(X)}$

Example: Toss two coins, X = number of tails.

X	Outcomes	P(X = x)
0	HH	1/4
1	HT, TH	1/2
2	TT	1/4

Discrete Probability Distributions

Bernoulli Distribution

The Bernoulli distribution models a single trial with two possible outcomes: "success" (with probability p) or "failure" (with probability 1-p).

• X = 1 if success, X = 0 if failure

• Probability mass function: $P(X = x) = p^x (1-p)^{1-x}, \; x = 0, 1$

Example: Inspecting a resistor for defects (p = 0.01):

• $P(X = 1) = 0.01$

• $P(X = 0) = 0.99$

Binomial Distribution

The binomial distribution models the number of successes in n independent Bernoulli trials, each with probability p of success.

• n independent trials

• Each trial has two outcomes: success or failure

• Probability of success is constant (p)

The probability of getting r successes in n trials is:

$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$

Expected value: $E(X) = np$

Variance: $Var(X) = np(1-p)$

Standard deviation: $SD(X) = \sqrt{np(1-p)}$

Examples:

• Probability of winning more than half of 30 games

• Probability that at least one of 10 wind turbines shuts down

• Probability that more than 4 out of 8 patients are cured by a drug

• Probability that 2 or more out of 5 people are left-handed (p = 0.10)

• Probability of more than one defective resistor in a sample of 20 (p = 0.05)

Poisson Distribution

The Poisson distribution models the number of independent events occurring in a fixed interval of time or space, given a constant average rate (λ).

• Events occur independently

• Probability of occurrence is constant over time

The probability of observing r events in an interval is:

$P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!}$

Where λ is the average number of events in the interval.

Examples:

• Number of calls at a call center per minute (λ = 3)

• Probability of exactly 2 calls in one minute

• Probability of exactly 10 calls in three minutes (λ = 9)

• Probability of at least one defective unit in a day (λ = 2)

Summary of Key Concepts

• For a discrete random variable, P(X = x) is the probability that X takes value x.

• Expected value (mean): $E(X) = \sum x_k P(X = x_k)$

• Variance: $Var(X) = E(X^2) - [E(X)]^2$

• Standard deviation: $SD(X) = \sqrt{Var(X)}$

• Binomial and Poisson distributions are important models for discrete random variables.