DTSA 5002 Statistical Inference & ML Exam LATEST
2026 UPDATE 100+ QUESTIONS AND DETAILED
VERIFIED ANSWERS FROM ACTUAL EXAMS TEST
GRADE A+
Section 1: Probability Foundations
Question 1
If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.4, what is P(A
∪ B)?
A) 0.12
B) 0.7
C) 0.58
D) 0.1
*Correct Answer: B — For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 0.3 +
0.4 = 0.7.*
Question 2
If P(A) = 0.6, P(B) = 0.5, and P(A ∩ B) = 0.3, what is P(A|B)?
A) 0.5
B) 0.6
C) 0.3
D) 0.83
*Correct Answer: B — P(A|B) = P(A∩B)/P(B) = 0.3/0.5 = 0.6.*
Question 3
Two fair dice are rolled. What is the probability of getting a sum of 7?
A) 1/6
B) 1/12
C) 5/36
D) 1/36
,*Correct Answer: A — There are 6 outcomes summing to 7 out of 36 total, so 6/36
= 1/6.*
Question 4
If events A and B are independent, P(A)=0.4, P(B)=0.5, then P(A∩B) equals:
A) 0.9
B) 0.2
C) 0.0
D) 0.45
*Correct Answer: B — For independent events, P(A∩B)=P(A)P(B)=0.4×0.5=0.2.*
Question 5
Bayes' theorem primarily allows us to:
A) Compute marginal probabilities from joint
B) Update prior probabilities with new evidence
C) Find the median of a distribution
D) Check independence
Correct Answer: B — Bayes' theorem reverses conditional probabilities to update
beliefs given data.
Question 6
A random variable X has E[X]=3, Var(X)=4. What is E[2X+1]?
A) 7
B) 6
C) 5
D) 4
*Correct Answer: A — E[2X+1]=2E[X]+1=2×3+1=7.*
Question 7
If X and Y are independent with Var(X)=2, Var(Y)=3, then Var(2X−Y) is:
A) 11
B) 7
C) 5
D) 8
,*Correct Answer: A — Var(2X−Y)=4Var(X)+Var(Y)=4×2+3=8+3=11.*
Question 8
A binomial random variable with n=10, p=0.3 has mean:
A) 3
B) 2.1
C) 7
D) 0.3
*Correct Answer: A — Mean of Binomial = n×p = 10×0.3 = 3.*
Question 9
The Poisson distribution is often used to model:
A) Number of successes in fixed trials
B) Waiting time until first event
C) Count of rare events in fixed interval
D) Continuous symmetric data
Correct Answer: C — Poisson models count of rare events per unit time/space.
Question 10
If Z ~ N(0,1), what is P(Z > 1.96) approximately?
A) 0.025
B) 0.05
C) 0.01
D) 0.10
Correct Answer: A — The upper tail probability for 1.96 in standard normal is
0.025.
Question 11
The central limit theorem states that the sample mean:
A) Is exactly normal for any n
B) Approaches normal distribution as n increases, regardless of population
distribution
C) Approaches the population mean in probability
D) Has variance equal to population variance
, Correct Answer: B — CLT says sum/mean of i.i.d. variables tends to normal, shape
irrelevant.
Question 12
Which of the following is a continuous distribution?
A) Binomial
B) Poisson
C) Normal
D) Geometric
Correct Answer: C — Normal is continuous; others are discrete.
Question 13
For a random sample from a population with mean μ, the expected value of the
sample mean is:
A) μ
B) σ/√n
C) μ²
D) 0
Correct Answer: A — E[X̄] = μ regardless of distribution.
Question 14
The variance of the sample mean (X̄) from i.i.d. draws with variance σ² and sample
size n is:
A) σ²
B) σ²/n
C) σ/√n
D) nσ²
Correct Answer: B — Var(X̄) = σ²/n.
Question 15
If two events are independent, then:
A) P(A∩B)=0
B) P(A|B)=P(A)
2026 UPDATE 100+ QUESTIONS AND DETAILED
VERIFIED ANSWERS FROM ACTUAL EXAMS TEST
GRADE A+
Section 1: Probability Foundations
Question 1
If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.4, what is P(A
∪ B)?
A) 0.12
B) 0.7
C) 0.58
D) 0.1
*Correct Answer: B — For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 0.3 +
0.4 = 0.7.*
Question 2
If P(A) = 0.6, P(B) = 0.5, and P(A ∩ B) = 0.3, what is P(A|B)?
A) 0.5
B) 0.6
C) 0.3
D) 0.83
*Correct Answer: B — P(A|B) = P(A∩B)/P(B) = 0.3/0.5 = 0.6.*
Question 3
Two fair dice are rolled. What is the probability of getting a sum of 7?
A) 1/6
B) 1/12
C) 5/36
D) 1/36
,*Correct Answer: A — There are 6 outcomes summing to 7 out of 36 total, so 6/36
= 1/6.*
Question 4
If events A and B are independent, P(A)=0.4, P(B)=0.5, then P(A∩B) equals:
A) 0.9
B) 0.2
C) 0.0
D) 0.45
*Correct Answer: B — For independent events, P(A∩B)=P(A)P(B)=0.4×0.5=0.2.*
Question 5
Bayes' theorem primarily allows us to:
A) Compute marginal probabilities from joint
B) Update prior probabilities with new evidence
C) Find the median of a distribution
D) Check independence
Correct Answer: B — Bayes' theorem reverses conditional probabilities to update
beliefs given data.
Question 6
A random variable X has E[X]=3, Var(X)=4. What is E[2X+1]?
A) 7
B) 6
C) 5
D) 4
*Correct Answer: A — E[2X+1]=2E[X]+1=2×3+1=7.*
Question 7
If X and Y are independent with Var(X)=2, Var(Y)=3, then Var(2X−Y) is:
A) 11
B) 7
C) 5
D) 8
,*Correct Answer: A — Var(2X−Y)=4Var(X)+Var(Y)=4×2+3=8+3=11.*
Question 8
A binomial random variable with n=10, p=0.3 has mean:
A) 3
B) 2.1
C) 7
D) 0.3
*Correct Answer: A — Mean of Binomial = n×p = 10×0.3 = 3.*
Question 9
The Poisson distribution is often used to model:
A) Number of successes in fixed trials
B) Waiting time until first event
C) Count of rare events in fixed interval
D) Continuous symmetric data
Correct Answer: C — Poisson models count of rare events per unit time/space.
Question 10
If Z ~ N(0,1), what is P(Z > 1.96) approximately?
A) 0.025
B) 0.05
C) 0.01
D) 0.10
Correct Answer: A — The upper tail probability for 1.96 in standard normal is
0.025.
Question 11
The central limit theorem states that the sample mean:
A) Is exactly normal for any n
B) Approaches normal distribution as n increases, regardless of population
distribution
C) Approaches the population mean in probability
D) Has variance equal to population variance
, Correct Answer: B — CLT says sum/mean of i.i.d. variables tends to normal, shape
irrelevant.
Question 12
Which of the following is a continuous distribution?
A) Binomial
B) Poisson
C) Normal
D) Geometric
Correct Answer: C — Normal is continuous; others are discrete.
Question 13
For a random sample from a population with mean μ, the expected value of the
sample mean is:
A) μ
B) σ/√n
C) μ²
D) 0
Correct Answer: A — E[X̄] = μ regardless of distribution.
Question 14
The variance of the sample mean (X̄) from i.i.d. draws with variance σ² and sample
size n is:
A) σ²
B) σ²/n
C) σ/√n
D) nσ²
Correct Answer: B — Var(X̄) = σ²/n.
Question 15
If two events are independent, then:
A) P(A∩B)=0
B) P(A|B)=P(A)
