PMF PMF Properties and Axioms

Question 1

Which two conditions must every valid PMF satisfy?

Accepted Answer

P(X=x) ≥ 0 for all x, and Σ P(X=x) = 1

Answer

A valid PMF requires all probabilities to be non-negative and all probabilities to sum to exactly 1.

Question 2

Which of the following is NOT a valid PMF for X ∈ {1, 2, 3}?

Accepted Answer

P(1)=0.4, P(2)=0.4, P(3)=0.3

Answer

P(1)+P(2)+P(3) = 0.4+0.4+0.3 = 1.1 ≠ 1, so this is not a valid PMF.

Question 3

What is the key difference between a PMF and a PDF (probability density function)?

Accepted Answer

PMF applies to discrete random variables; PDF applies to continuous random variables

Answer

PMF gives exact probabilities for discrete outcomes; PDF gives probability density for continuous variables (probabilities require integration).

Question 4

For a valid PMF, can P(X = x₀) = 0 for some specific value x₀ in the support?

Accepted Answer

Yes, a PMF can assign zero probability to values in its defined domain

Answer

P(X=x) ≥ 0 allows values of zero; only values in the support (where P > 0) are typically listed.

Question 5

What is the support of a PMF?

Accepted Answer

The set of all values x where P(X=x) > 0

Answer

The support is the set of values that have positive probability under the PMF.

PMF - Probability Mass Function Practice Test