SL 4.7 — Discrete Random Variables

Content	Guidance, Clarification & Links
• Concept of discrete random variables • Probability distributions • Expected value (mean) • Applications (fairness, games, decisions)	• Tables and formulas may be used for probability distributions • Expected value used to evaluate fairness • Probabilities must sum to 1 • Applications include gambling, business risk & modelling

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Understanding Discrete Random Variables

A discrete random variable (DRV) is a variable that takes a list of countable outcomes.
Examples include:

• Dice results {1,2,3,4,5,6}
• Number of goals scored
• Number of defective items

A DRV is paired with a probability distribution describing how likely each value is.

Probability Distributions (Table Form)

X	1	2	3	4	5
P(X=x)	0.1	0.2	0.15	0.05	0.5

This is a valid probability distribution because:

• All probabilities are ≥ 0
• The total = 1

Probability Distributions (Function Form)

Sometimes probabilities are expressed as a rule:

P(X = x) = (1/18)(4 + x) for x ∈ {1,2,3}

You must verify:

• Probabilities are positive
• The sum equals 1

Expected Value (Mean)

The expected value is the long-run average:

E(X) = Σ[x × P(X = x)]

Example

Using the table above:

E(X) = 1(0.1) + 2(0.2) + 3(0.15) + 4(0.05) + 5(0.5)
= 3.65

Applications (Fair Games, Risk Analysis)

Expected value helps evaluate:

• Fairness of games (E(X) = 0 → fair game)
• Casino profitability (house edge)
• Insurance calculations
• Decision-making under uncertainty