PMF Hypergeometric Distribution 1 — Questions and Answers
Question 1: When does the Hypergeometric distribution apply instead of the Binomial?
- When sampling is done without replacement from a finite population (Correct answer)
- When sampling is done with replacement
- When events are independent
- When the population is infinite
Correct answer: When sampling is done without replacement from a finite population
The Hypergeometric distribution applies when drawing without replacement from a finite population, making draws dependent.
Question 2: A box has 5 red and 7 blue balls. You draw 4 without replacement. What distribution models the number of red balls drawn?
- Hypergeometric (Correct answer)
- Binomial
- Poisson
- Geometric
Correct answer: Hypergeometric
Sampling without replacement from a finite population of two types is modeled by the Hypergeometric distribution.
Question 3: What are the parameters of the Hypergeometric distribution?
- N (population size), K (successes in population), n (sample size) (Correct answer)
- n (trials), p (success probability)
- λ (rate), k (count)
- N (population), p (proportion), n (sample)
Correct answer: N (population size), K (successes in population), n (sample size)
The Hypergeometric distribution is parameterized by N (total population), K (number of successes in population), and n (sample size).
Question 4: What is the PMF of the Hypergeometric distribution?
- P(X=k) = C(K,k)×C(N-K,n-k) / C(N,n) (Correct answer)
- P(X=k) = C(n,k) × p^k × (1-p)^(n-k)
- P(X=k) = e^-λ × λ^k / k!
- P(X=k) = (1-p)^(k-1) × p
Correct answer: P(X=k) = C(K,k)×C(N-K,n-k) / C(N,n)
The Hypergeometric PMF uses combinations: C(K,k) ways to choose k successes times C(N-K, n-k) ways to choose remaining items, divided by C(N,n) total ways.
Question 5: What is the mean of a Hypergeometric distribution with parameters N, K, n?
- n × K/N (Correct answer)
- K/N
- n × p
- nK
Correct answer: n × K/N
E[X] = n × K/N, which is the sample size times the population proportion of successes.
Question 6: When does the Hypergeometric distribution closely approximate the Binomial?
- When the sample size n is much smaller than the population N (Correct answer)
- When n equals N
- When K is very small
- When p is close to 0.5
Correct answer: When the sample size n is much smaller than the population N
When n is much smaller than N (n/N is small), sampling without replacement approximates sampling with replacement, and Hypergeometric ≈ Binomial.
PMF Hypergeometric Distribution 1
When does the Hypergeometric distribution apply instead of the Binomial?