Discrete Random Variables

Random Variable

• A numerical variable whose value depends on the outcome of a random process.

• Gets denoted by uppercase letters (X, Y)

Discrete Random Variable

A variable that can take a finite or countably infinite set of possible values.

Probability Mass Function (PMF)

• Function that gives the probability that a discrete random variable equals a specific value

• P(X = x) = p(x)

• where Σp(x) = 1

Properties of a PMF

• 0 ≤ p(x) ≤ 1 for all x

• Σp(x) = 1

Cumulative Distribution Function (CDF)

• F(x) = P(X ≤ x) = Σ{t ≤ x}

• Non-decreasing

• Approaches 1 as x → ∞

X

Symbol that represents the set of all x values where p(x) > 0

p(x) = 1/6

What will be the PMF of a fair sided die.

x = 1, 2, 3, 4, 5, 6.

x values of the fair sides die

Indicator variable

A discrete random variable that takes value 1 if an event occurs and 0 otherwise.

Joint PMF

• For two discrete variables X and Y

• p(x, y) = P(X = x, Y = y)

Marginal PMF

Found by summing joint probabilities:

• p_X (x) = Σ_y p(x, y)

• p_Y (y) = Σ_x p(x, y)

Conditional PMF

• p_{Y|X} (y|x) = p(x, y) / p_X (x)

• if p_X (x) > 0