– Spring 2025 Complete Study Guide
with Verified Questions, Answers &
Rationales.
Section 1: Probability and Statistics Foundations
Question 1
What is the probability of rolling a sum of 7 with two fair six-sided dice?
A. 1/36
B. 5/36
C. 6/36 = 1/6
D. 1/2
Correct Answer: C
Explanation: There are 6 combinations that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
out of 36 total possible outcomes. 6/36 simplifies to 1/6.
Question 2
If P(A) = 0.3, P(B) = 0.4, and A and B are independent, what is P(A ∪ B)?
A. 0.12
B. 0.58
C. 0.70
D. 0.82
Correct Answer: B
Explanation: For independent events, P(A ∩ B) = P(A)P(B) = 0.3 × 0.4 = 0.12. Then P(A ∪
B) = P(A) + P(B) - P(A ∩ B) = 0.3 + 0.4 - 0.12 = 0.58.
Question 3
What is the value of the standard normal PDF φ(z) at z = 0?
, A. 0
B. 0.3989
C. 0.5000
D. 1.0000
Correct Answer: B
Explanation: φ(0) = 1/√(2π) ≈ 0.3989. This is the maximum value of the standard
normal probability density function.
Question 4
For a standard normal random variable Z, what is P(Z > 1.96)?
A. 0.025
B. 0.05
C. 0.95
D. 0.975
Correct Answer: A
Explanation: From the standard normal table, P(Z > 1.96) = 0.025. This is a critical value
commonly used for 95% confidence intervals.
Question 5
What is the expected value of a Bernoulli(p) random variable?
A. 0
B. p
C. 1-p
D. 1
Correct Answer: B
Explanation: For a Bernoulli(p) random variable X, P(X=1)=p and P(X=0)=1-p. So E[X] =
1×p + 0×(1-p) = p.
Question 6
What is the variance of a Bernoulli(p) random variable?
A. p
B. 1-p
C. p(1-p)
D. p²
, Correct Answer: C
Explanation: Var(X) = E[X²] - (E[X])² = p - p² = p(1-p). This is maximum at p=0.5.
Question 7
If X ~ Binomial(n,p), what is E[X]?
A. np
B. np(1-p)
C. p
D. n
Correct Answer: A
Explanation: The binomial distribution is the sum of n independent Bernoulli(p) random
variables. By linearity of expectation, E[X] = np.
Question 8
If X ~ Binomial(n,p), what is Var(X)?
A. np
B. np(1-p)
C. p(1-p)
D. n²p(1-p)
Correct Answer: B
Explanation: The variance of the sum of independent random variables is the sum of
the variances. For n i.i.d. Bernoulli(p), Var(X) = np(1-p).
Question 9
A random variable with equal mean and variance is most likely which distribution?
A. Normal
B. Binomial
C. Poisson
D. Uniform
Correct Answer: C
Explanation: For a Poisson distribution with parameter λ, both the mean and variance
are equal to λ.
Question 10
What is the probability that a Poisson(λ=4) random variable equals 3?
, A. 0.0733
B. 0.1465
C. 0.1954
D. 0.2240
Correct Answer: C
Explanation: P(X=3) = e⁻⁴ × 4³ / 3! = e⁻⁴ × = e⁻⁴ × 10.6667 ≈ 0.0183 × 10.6667 =
0.1954.
Question 11
For the exponential distribution with rate λ, what is the mean?
A. λ
B. 1/λ
C. λ²
D. 1/λ²
Correct Answer: B
Explanation: The exponential distribution has mean 1/λ and variance 1/λ². The rate
parameter λ is the reciprocal of the mean.
Question 12
The memoryless property of the exponential distribution states:
A. P(X > s + t) = P(X > s) + P(X > t)
B. P(X > s + t | X > s) = P(X > t)
C. P(X > s + t) = P(X > s)P(X > t)
D. P(X < s + t | X > s) = P(X < t)
Correct Answer: B
Explanation: The exponential distribution is the only continuous distribution with the
memoryless property: the remaining lifetime does not depend on the current age.
Question 13
If X and Y are independent random variables, then Cov(X,Y) is:
A. 1
B. 0
C. Var(X)
D. Var(Y)