ISYE 6644 OH EXAM 1 PRACTICE QUESTIONS – SPRING
2026 Complete Study Guide with 130 Verified Questions,
Answers & Detailed Rationales Updated for 2025/2026 |
100% Verified | Pass Guaranteed – A+ Graded | Instant
Download PDF
CORE DOMAINS COVERED
1. Probability & Statistics Foundations - Basic probability, random variables,
distributions, and moments
2. Discrete Distributions - Bernoulli, Binomial, Geometric, Negative Binomial,
Poisson, and Hypergeometric
3. Continuous Distributions - Uniform, Exponential, Normal, Gamma, Weibull, and
Beta distributions
4. Multivariate Distributions - Joint distributions, covariance, correlation, and
conditional probability
5. Expectation & Variance - Properties, functions of random variables, and moment
generating functions
6. Normal Distribution & CLT - Standard normal, z-scores, Central Limit Theorem,
and sampling distributions
7. Data Analysis & Descriptive Statistics - Summary statistics, histograms, and data
visualization
8. Statistical Inference & Estimation - Point estimation, confidence intervals, and
hypothesis testing
SECTION 1: PROBABILITY & STATISTICS FOUNDATIONS (Questions 1-25)
Q1. 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
Rationale: 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.
Q2. 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
Rationale: 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.
Q3. 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
Rationale: φ(0) = 1/√(2π) ≈ 0.3989. This is the maximum value of the standard normal
probability density function.
Q4. 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
,Rationale: From the standard normal table, P(Z > 1.96) = 0.025. This is a critical value
commonly used for 95% confidence intervals.
Q5. What is the expected value of a Bernoulli(p) random variable?
A. 0
B. p
C. 1-p
D. 1
Rationale: 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.
Q6. What is the variance of a Bernoulli(p) random variable?
A. p
B. 1-p
C. p(1-p)
D. p²
Rationale: Var(X) = E[X²] - (E[X])² = p - p² = p(1-p). This is the standard variance formula for
a Bernoulli random variable.
Q7. If X and Y are independent random variables, what is Var(X + Y)?
A. Var(X) + Var(Y)
B. Var(X) - Var(Y)
C. Var(X) + Var(Y) + 2Cov(X,Y)
D. Var(X) + Var(Y) - 2Cov(X,Y)
A. Var(X) + Var(Y)
Rationale: For independent random variables, Cov(X,Y) = 0, so Var(X + Y) = Var(X) +
Var(Y) + 2Cov(X,Y) = Var(X) + Var(Y).
, Q8. What is the probability of getting exactly 3 heads in 5 tosses of a fair coin?
A. 5/32
B. 10/32 = 5/16
C. 15/32
D. 20/32
Rationale: This is a binomial problem: C(5,3) × (1/2)³ × (1/2)² = 10 × 1/32 = 10/32 = 5/16.
Q9. If P(A) = 0.6 and P(B|A) = 0.2, what is P(A ∩ B)?
A. 0.08
B. 0.12
C. 0.30
D. 0.80
Rationale: P(A ∩ B) = P(A) × P(B|A) = 0.6 × 0.2 = 0.12.
Q10. What is the coefficient of variation for a random variable with mean μ and
standard deviation σ?
A. μ/σ
B. σ/μ
C. μ/σ²
D. σ²/μ
Rationale: The coefficient of variation (CV) is defined as σ/μ, representing the relative
variability of the random variable.
Q11. Which of the following is NOT a valid probability?
A. 0
B. 0.5
C. 1
D. 1.5
2026 Complete Study Guide with 130 Verified Questions,
Answers & Detailed Rationales Updated for 2025/2026 |
100% Verified | Pass Guaranteed – A+ Graded | Instant
Download PDF
CORE DOMAINS COVERED
1. Probability & Statistics Foundations - Basic probability, random variables,
distributions, and moments
2. Discrete Distributions - Bernoulli, Binomial, Geometric, Negative Binomial,
Poisson, and Hypergeometric
3. Continuous Distributions - Uniform, Exponential, Normal, Gamma, Weibull, and
Beta distributions
4. Multivariate Distributions - Joint distributions, covariance, correlation, and
conditional probability
5. Expectation & Variance - Properties, functions of random variables, and moment
generating functions
6. Normal Distribution & CLT - Standard normal, z-scores, Central Limit Theorem,
and sampling distributions
7. Data Analysis & Descriptive Statistics - Summary statistics, histograms, and data
visualization
8. Statistical Inference & Estimation - Point estimation, confidence intervals, and
hypothesis testing
SECTION 1: PROBABILITY & STATISTICS FOUNDATIONS (Questions 1-25)
Q1. 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
Rationale: 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.
Q2. 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
Rationale: 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.
Q3. 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
Rationale: φ(0) = 1/√(2π) ≈ 0.3989. This is the maximum value of the standard normal
probability density function.
Q4. 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
,Rationale: From the standard normal table, P(Z > 1.96) = 0.025. This is a critical value
commonly used for 95% confidence intervals.
Q5. What is the expected value of a Bernoulli(p) random variable?
A. 0
B. p
C. 1-p
D. 1
Rationale: 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.
Q6. What is the variance of a Bernoulli(p) random variable?
A. p
B. 1-p
C. p(1-p)
D. p²
Rationale: Var(X) = E[X²] - (E[X])² = p - p² = p(1-p). This is the standard variance formula for
a Bernoulli random variable.
Q7. If X and Y are independent random variables, what is Var(X + Y)?
A. Var(X) + Var(Y)
B. Var(X) - Var(Y)
C. Var(X) + Var(Y) + 2Cov(X,Y)
D. Var(X) + Var(Y) - 2Cov(X,Y)
A. Var(X) + Var(Y)
Rationale: For independent random variables, Cov(X,Y) = 0, so Var(X + Y) = Var(X) +
Var(Y) + 2Cov(X,Y) = Var(X) + Var(Y).
, Q8. What is the probability of getting exactly 3 heads in 5 tosses of a fair coin?
A. 5/32
B. 10/32 = 5/16
C. 15/32
D. 20/32
Rationale: This is a binomial problem: C(5,3) × (1/2)³ × (1/2)² = 10 × 1/32 = 10/32 = 5/16.
Q9. If P(A) = 0.6 and P(B|A) = 0.2, what is P(A ∩ B)?
A. 0.08
B. 0.12
C. 0.30
D. 0.80
Rationale: P(A ∩ B) = P(A) × P(B|A) = 0.6 × 0.2 = 0.12.
Q10. What is the coefficient of variation for a random variable with mean μ and
standard deviation σ?
A. μ/σ
B. σ/μ
C. μ/σ²
D. σ²/μ
Rationale: The coefficient of variation (CV) is defined as σ/μ, representing the relative
variability of the random variable.
Q11. Which of the following is NOT a valid probability?
A. 0
B. 0.5
C. 1
D. 1.5
