Combinatorics • Topic 23

Random Variables

A random variable is a variable whose value is subject to variations due to chance. It bridges the gap between abstract outcomes and concrete numbers.

Definition

Definition (Random Variable)

A random variable

X

is a function that assigns a real number to each outcome in the sample space

S

X : S \to R

Discrete Random Variables

X

can take on a countable set of distinct values (e.g., integers).

Probability Mass Function (PMF): $P (X = k)$ .

all k \sum P (X = k) = 1

Common Distributions

1. Bernoulli( $p$ )

Success (1) with prob

p

, Failure (0) with prob

1 - p

Example: One coin flip.

2. Binomial( $n, p$ )

Number of successes in

n

independent Bernoulli trials.

P (X = k) = (k n) p^{k} (1 - p)^{n - k}

3. Geometric( $p$ )

Number of trials needed to get the first success.

P (X = k) = (1 - p)^{k - 1} p

Practice Problems

Exercise (Problem 1)

Let

X

be the sum of two fair dice. Write out the PMF table for

X

Exercise (Problem 2)

A fair coin is flipped 5 times. What is the probability that the number of heads

X

is exactly 3?

Exercise (Problem 3)

Let

X

be the number of distinct values obtained when rolling 3 dice. Find

P (X = 2)