Questions: Random Variables

Question 1

You flip a fair coin three times and define X as the number of heads. What kind of object is X?

• A. A single probability value, since each flip has a 50% chance of heads
• B. A random process, since the outcome changes each time you flip
• C. A function from the sample space {HHH, HHT, HTH, ...} to the real numbers {0, 1, 2, 3}
• D. An event, specifically the event that heads occurs

Answer: (C)

A random variable is formally a function X: Ω → ℝ. The sample space Ω contains all 8 outcomes, and X assigns a number to each: X(HHH) = 3, X(HHT) = 2, X(HTH) = 2, etc. X is not itself random — it's a deterministic function. The randomness comes from the uncertain input (which outcome actually occurs). Options A and D confuse a random variable with a probability or an event.

Question 2

Two random variables X and Y are defined on completely different sample spaces but have identical distributions. Which of the following must be true?

• A. X and Y must produce the same numerical values in the same order when the experiments are run simultaneously
• B. X and Y behave identically in every probabilistic sense — any probability statement about one holds for the other
• C. X and Y are really the same random variable, just described differently
• D. X and Y have the same expected value but may differ in variance

Answer: (B)

Two random variables with the same distribution are probabilistically identical: P(X ∈ A) = P(Y ∈ A) for every set A. Their underlying sample spaces may look completely different, but if the distribution is the same, all probabilistic computations agree. The distribution is the complete probabilistic summary of a random variable. Option D is wrong — identical distributions implies identical expected value AND variance AND all other distributional properties.

Question 3

A random variable is literally a variable whose value randomly changes over time.

• T. True
• F. False

Answer: False

A random variable is a function, not a 'variable' in the sense of something changing over time. It is a fixed, deterministic mapping from outcomes in a sample space to real numbers. The 'randomness' refers to uncertainty about which input (outcome) will occur, not to the function itself changing. When an experiment is performed, a specific outcome is realized and the random variable maps it to a specific number — the function is deterministic; only the input is uncertain.

Question 4

The probabilities in a discrete random variable's distribution must sum to 1 because this follows from the probability axioms applied to the underlying sample space.

• T. True
• F. False

Answer: True

The events {X = x} for all possible values x form a partition of the sample space Ω — every outcome maps to exactly one value of X. By the probability axioms, probabilities of a partition sum to 1. So the distribution of X automatically satisfies this property: P(X = x₁) + P(X = x₂) + ... = 1. This is not an additional requirement imposed on random variables — it follows directly from the structure of the underlying probability space.

Question 5

Why do we introduce random variables rather than working directly with events and probabilities? What does the numerical structure add?

Model answer: Random variables translate abstract outcomes into numbers, enabling arithmetic operations that have no analogue for non-numerical events. Once outcomes become numbers, we can compute averages (expected value), measure spread (variance), add two random variables, and apply the full toolkit of real analysis. Working with raw events like 'heads occurred' doesn't support these operations. Random variables also enable comparison: two experiments with different sample spaces can have random variables with the same distribution, making their probabilistic behavior directly comparable even though their underlying outcomes look nothing alike.

The key insight is abstraction: instead of studying each probability experiment from scratch, we study distributions — and distributions live in a common mathematical space (ℝ) where we have powerful tools. This is why random variables become the standard language of probability from this point forward.