L7 Discrete Random Variables and Probability Distributions: Study Notes

Random Variables and Probability Distributions

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 statistics for quantifying outcomes and analyzing probability.

• Definition: A random variable, denoted as X, maps outcomes of an experiment to numbers.

• Example: Tossing 3 coins, let X be the number of tails. Possible values: 0, 1, 2, or 3.

• Notation: P(X = x) is the probability that X takes the value x.

Probability Distribution of a Random Variable

The probability distribution of a random variable describes how probabilities are assigned to each possible value of the variable.

• Probabilities must sum to 1 (100%).

• Example: Tossing 3 coins, probability distribution for number of tails:

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

Types of Random Variables

Random variables are classified as either discrete or continuous:

• Discrete Random Variables: Take on countable values (e.g., 0, 1, 2, ...).

• Continuous Random Variables: Can take any value within an interval.

Examples of discrete random variables:

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

• Number of forms with errors in a box of 20 (X = 0, 1, 2, ... 20)

• Number of sales from 10 cold calls (X = 0, 1, 2, ... 10)

Expected Value, Variance, and Standard Deviation

The expected value (mean), variance, and standard deviation are key measures for random variables:

• Expected Value (Mean): $E(X) = \sum_x x P(X = x)$

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

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

Example: Fire insurance payout: $X$ is either 0 (no fire) or 200,000 (fire). If $P(X = 200,000) = 1/10,000$ and $P(X = 0) = 9,999/10,000$, then:

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

• Insurance premium charged is much greater than $20$ to ensure profitability.

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 an experiment with two possible outcomes: success or failure.

• Definition: One trial, two outcomes. $P_{success} = p$.

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

• Example: Inspecting a resistor: $p = 0.01$ (defective), $X = 1$ if defective, $X = 0$ otherwise.

• $P(X = 1) = 0.01$, $P(X = 0) = 0.99$

Binomial Distribution

The binomial distribution models the number of successes in $n$ independent trials, each with two possible outcomes.

• Properties:

  • $n$ independent trials

  • Each trial has two outcomes: "success" or "failure"

  • Probability of success $p$ is constant

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

• Expected value: $E(X) = np$

• Variance: $Var(X) = np(1 - p)$

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

Examples:

• Chance of winning more than half of 30 games

• Probability that one wind turbine out of 10 shuts down

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

• Probability that none, one, or two or more are left-handed in a sample of 5 (with $p = 0.10$)

• Probability of receiving more than one defective resistor out of 20 (with $p = 0.05$)

Poisson Distribution

The Poisson distribution models the number of independent events occurring in a fixed interval of time, given a constant average rate $\lambda$.

• Properties:

  • Events occur independently

  • Probability of occurrence is constant over time

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

• Mean: $\lambda$

• Variance: $\lambda$

Examples:

• Number of traffic accidents per month at an intersection

• Number of website visitors per minute

• Number of patients arriving at A&E in 30 minutes

• Probability of exactly 2 calls in one minute (with $\lambda = 3$)

• Probability of exactly 10 calls in 3 minutes (with $\lambda = 9$)

• Probability of at least one defective unit in a day (with $\lambda = 2$)

Summary of Key Concepts

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

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

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

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

• Binomial and Poisson distributions are important discrete probability distributions.