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Please show your work before checking the answer and explanations. The exam questions will be similar to these exercise problems.
Problem 1. A biased coin has a 70% chance of landing heads. If X = 1 for heads and X = 0 for tails, what is P(X = 0)?
a) 0.3
b) 0.7
c) 0.49
d) 0.21
View Answer
Ans : a) 0.3
Explanation: For Bernoulli(p), P(X=0) = 1 - p. Here p = 0.7, so P(X=0) = 0.3.
Problem 2. Given a Bernoulli distribution with p = 0.6, what are its mean (μ) and variance (σ²)?
a) μ=0.6, σ²=0.24
b) μ=0.6, σ²=0.4
c) μ=0.4, σ²=0.24
d) μ=0.4, σ²=0.6
View Answer
Ans : a) μ=0.6, σ²=0.24
Explanation: For Bernoulli(p): Mean = p = 0.6. Variance = p(1-p) = 0.6 * 0.4 = 0.24.
Problem 3. Which of the following is a Bernoulli trial?
a) Rolling a die and recording the outcome
b) Flipping a coin once and recording heads/tails
c) Drawing a card from a deck
d) Measuring the height of a person
View Answer
Ans : b) Flipping a coin once and recording heads/tails
Explanation: A Bernoulli trial has exactly two mutually exclusive outcomes (success/failure) for a single event.
Problem 4. The probability mass function for a Binomial(n,p) distribution using factorial notation is:
a) P(X=k) = [n!/(k!(n-k)!)] * p^k * (1-p)^(n-k)
b) P(X=k) = [n!/(k!(n-k)!)] * p^(n-k) * (1-p)^k
c) P(X=k) = p^k * (1-p)^(n-k)
d) P(X=k) = (λ^k * e^(-λ))/k!
View Answer
Ans : a) P(X=k) = [n!/(k!(n-k)!)] * p^k * (1-p)^(n-k)
Explanation: This is the standard Binomial PMF formula with factorial notation for combinations.
Problem 5. A student guesses on a 10-question true/false quiz (p=0.5). What is P(exactly 8 correct)?
a) 0.0439
b) 0.1209
c) 0.0440
d) 0.0107
View Answer
Ans : a) 0.0439
Explanation: n=10, k=8, p=0.5. P(X=8) = [10!/(8!2!)](0.5)^8(0.5)^2 = 45(0.00390625)(0.25) ≈ 0.0439.
Problem 6. In 20 independent trials with success probability 0.3, what is the expected number of successes?
a) 6
b) 10
c) 4.2
d) 14
View Answer
Ans : a) 6
Explanation: For Binomial(n,p), mean = n*p = 20*0.3 = 6.
Problem 7. For a Binomial(15, 0.4) distribution, the variance is:
a) 6
b) 3.6
c) 9
d) 2.4
View Answer
Ans : b) 3.6
Explanation: Variance = n*p*(1-p) = 15*0.4*0.6 = 3.6
Problem 8. Which scenario is NOT well-modeled by a binomial distribution?
a) Number of defective items in a batch of 50
b) Number of heads in 100 coin flips
c) Number of emails you receive in an hour
d) Number of correct answers on a multiple-choice test (guessing)
View Answer
Ans : c) Number of emails you receive in an hou
Explanation: Emails per hour is better modeled by Poisson distribution (events over time). Binomial requires fixed number of trials.
Problem 9. If X ~ Binomial(n=5, p=0.2), what is P(X ≤ 2) expressed using the binomial PMF?
a) Σ[k=0 to 2] [5!/(k!(5-k)!)] * (0.2)^k * (0.8)^(5-k)
b) 1 - [5!/(3!2!)](0.2)^3(0.8)^2
c) [5!/(2!3!)] * (0.2)^2 * (0.8)^3
d) 1 - Σ[k=3 to 5] [5!/(k!(5-k)!)] * (0.2)^k * (0.8)^(5-k)
View Answer
Ans : a) Σ[k=0 to 2] [5!/(k!(5-k)!)] * (0.2)^k * (0.8)^(5-k)
Explanation: P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2), each calculated using the binomial PMF with factorial notation.
Problem 10. The probability mass function for Poisson(λ) is:
a) P(X=k) = (λ^k * e^(-λ))/k!
b) P(X=k) = [n!/(k!(n-k)!)] * p^k * (1-p)^(n-k)
c) P(X=k) = p*(1-p)^(k-1)
d) P(X=k) = e^(-λ) * λ^k
View Answer
Ans : D
Explanation: For a continuous distribution, the probability of selecting any arbitrary number is ALWAYS ZERO.
Problem 11. Accidents occur at an intersection at an average rate of 2 per month. What is P(exactly 3 accidents in a month)?
a) (e^(-2) * 2^3)/3!
b) (e^(-3) * 3^2)/2!
c) C(2,3) * (1/3)^3 * (2/3)^(-1)
d) 2^3/3!
View Answer
Ans : a) (e^(-2) * 2^3)/3!
Explanation: λ=2, k=3. Using Poisson PMF: P(X=3) = (e^(-2) * 2^3)/3!
Problem 12. If X follows Poisson distribution with mean 4.5, what is its variance?
a) 4.5
b) 2.25
c) 20.25
d) 9
View Answer
Ans : a) 4.5
Explanation: For Poisson distribution, mean = variance = λ.
Problem 13. The Poisson distribution is often used to approximate the Binomial distribution when:
a) n is large and p is close to 0.5
b) n is small and p is large
c) n is large and p is small
d) n is small and p is small
View Answer
Ans : c) n is large and p is small
Explanation: Poisson approximation works when n ≥ 20 and p ≤ 0.05, or n ≥ 100 and np ≤ 10.
Problem 14. If calls arrive at a rate of 1.5 per minute, what is P(no calls in a minute)?
a) e^(-1.5)
b) 1 - e^(-1.5)
c) 1.5 * e^(-1.5)
d) 0
View Answer
Ans : a) e^(-1.5)
Explanation: For Poisson(λ), P(X=0) = e^(-λ) = e^(-1.5).
Problem 15. For X ~ Binomial(n=10, p=0.5), what is the approximate 75th percentile? Given: P(X≤6) = Σ[k=0 to 6] [10!/(k!(10-k)!)](0.5)^k(0.5)^(10-k) ≈ 0.828
a) 6
b) 7
c) 8
d) 5
View Answer
Ans : a) 6
Explanation: The 75th percentile is the smallest x where P(X ≤ x) ≥ 0.75. P(X≤6)≈0.828 ≥ 0.75, and P(X≤5)≈0.623 < 0.75, so 6 is the 75th percentile.
Problem 16. Which distribution has variance equal to mean?
a) Binomial only
b) Poisson only
c) Both Binomial and Poisson
d) Bernoulli only
View Answer
Ans : b) Poisson only
Explanation: For Poisson: mean = variance = λ. For Binomial: mean = np, variance = np(1-p) which is less than mean unless p=0.
Problem 17. If X ~ Poisson(λ=3), find the smallest k such that P(X ≤ k) ≥ 0.9. Given: P(X≤4)≈0.815, P(X≤5)≈0.916.
a) 4
b) 5
c) 6
d) 3
View Answer
Ans : b) 5
Explanation: P(X≤4)=0.815 < 0.9, P(X≤5)=0.916 ≥ 0.9, so k=5 is the 90th percentile.
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