PMF Cheat Sheet 2026

The 30 highest-yield PMF facts, distilled from real exam questions. Print it, save it as a PDF, or study it here — free, no sign-up.

1. Which property makes the Geometric distribution unique among discrete distributions? → Memoryless property
2. What is the mode of a PMF? → The value x that maximizes P(X=x)
3. The Negative Binomial distribution is overdispersed relative to the Poisson. What does this mean? → Its variance exceeds its mean
4. A call center receives an average of 5 calls per hour. What distribution models the number of calls in any given hour? → Poisson distribution
5. A random variable which takes values greater than or equal to zero with probability one is... → Nonnegative
6. What is the support of the Negative Binomial (total trials until r-th success)? → {r, r+1, r+2, …}
7. What is the relationship between E[X] and the median of a PMF? → They are equal only when the distribution is symmetric
8. For a fair six-sided die, what is E[X]? → 3.5
9. A hypergeometric scenario has N=10, K=6, n=4. What is the maximum possible value of X? → 4
10. If X ~ Poisson(λ=2), what is P(X=0)? → e^-2 ≈ 0.135
11. All of the following are binomial experiments except: → Rolling a die until a 6 appears
12. What is the second moment E[X²] used for in PMF calculations? → Computing variance via Var(X) = E[X²] - (E[X])²
13. What property of the CDF makes it easy to compute P(a < X ≤ b)? → P(a < X ≤ b) = F(b) - F(a)
14. For a Geometric distribution (number of failures before first success), what is the PMF? → P(X=k) = (1-p)^k × p
15. For a joint PMF p(x, y), the covariance Cov(X, Y) is computed as: → E[XY] − E[X]·E[Y]
16. Which of these restricts the measurement of a continuous variable? → Accuracy
17. A box has 5 red and 7 blue balls. You draw 4 without replacement. What distribution models the number of red balls drawn? → Hypergeometric
18. The finite population correction factor in the Hypergeometric variance is (N-n)/(N-1). What happens to this factor as n approaches N? → It approaches 0, reducing the variance
19. What is the mean of a Negative Binomial distribution (failures before r-th success)? → r(1-p)/p
20. In a traffic light consisting of red, yellow and green. What is the probability of getting 2 reds? → 0.189
21. Can a PMF have more than one mode? → Yes, if two or more values share the maximum probability
22. For the Negative Binomial (total trials until r-th success), what is the PMF? → P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r)
23. A card is drawn from a deck of 52 without replacement for 5 draws. Which distribution models the number of aces drawn? → Hypergeometric (N=52, K=4, n=5)
24. If the Poisson parameter λ increases, how does the shape of the PMF change? → It shifts right and becomes more symmetric
25. For a Negative Binomial with r=2 and p=0.5, what is E[X] (failures before 2nd success)? → 2
26. What does the Geometric distribution model? → The number of trials until the first success
27. For a valid PMF, can P(X = x₀) = 0 for some specific value x₀ in the support? → Yes, a PMF can assign zero probability to values in its defined domain
28. How is the CDF F(x) related to the PMF P(X=k) for discrete random variables? → F(x) = Σ P(X=k) for all k ≤ x
29. Given joint PMF p(1,1)=0.1, p(1,2)=0.2, p(2,1)=0.3, p(2,2)=0.4, what is the marginal PMF p_X(1)? → 0.3
30. For a Poisson PMF, which of the following is true about its support (possible values)? → X can take any non-negative integer value (0, 1, 2, …)