A fair six-sided die has PMF p(x) = 1/6 for x ∈ {1,2,3,4,5,6}. What is P(X ≥ 5)?
A1/6
B1/3
C1/2
D5/6
P(X ≥ 5) = P(X = 5) + P(X = 6) = 1/6 + 1/6 = 2/6 = 1/3. For discrete variables, probabilities of ranges are computed by summing the PMF over the relevant values — unlike continuous distributions where you integrate a density.
Question 2 True / False
For a continuous random variable (like human height), the probability mass function gives the probability that the variable equals exactly any specific value.
TTrue
FFalse
Answer: False
PMFs only apply to discrete random variables — those that take countable values. Continuous random variables use a probability density function (PDF), where P(X = any exact value) = 0. The PDF gives density, not probability; you must integrate over an interval to get a probability.
Question 3 Short Answer
A PMF assigns probability 0.4 to x = 0, probability 0.35 to x = 1, and probability 0.25 to x = 2. Explain why this is a valid PMF.
Think about your answer, then reveal below.
Model answer: It is valid because all values are non-negative (0.4, 0.35, 0.25 ≥ 0) and they sum to exactly 1 (0.4 + 0.35 + 0.25 = 1.00). These are the two requirements for any valid PMF.
The two axioms of a PMF mirror the probability axioms: probabilities cannot be negative, and the total probability across all possible outcomes must equal 1. Any function satisfying these two conditions over a discrete domain is a valid PMF.