Probability questions in GATE CSE are short but need the right property or distribution at once. This GATE-style set covers random variables, rules for expectation and variance, the standard discrete and continuous distributions, conditional expectation, covariance and correlation, basic statistics and two-state Markov chains.
How to practise: read each question and tap the option you think is correct. You will see right away whether you got it, along with a short explanation, and your score is tracked at the bottom of the screen. This set has 15 questions.
Topics covered in this set
- Random variables and expectation
- Properties of variance
- Binomial, Poisson and geometric distributions
- Exponential and normal distributions
- Conditional expectation, covariance and correlation
- Mean, median, mode and Markov chains
GATE CSE Engineering Mathematics MCQs with answers and explanations
Q1. A discrete random variable X has P(X = x) = kx for x = 1, 2, 3, 4 and 0 otherwise. What is E[X]?
a) 3
b) 5/2
c) 10/3
d) 2
Q2. A random variable X has variance 4. What is Var(2X + 3)?
a) 16
b) 8
c) 19
d) 11
Q3. X and Y are independent random variables with Var(X) = 3 and Var(Y) = 5. What is Var(X − Y)?
a) 8
b) −2
c) 15
d) 2
Q4. A binomial random variable has mean 4 and variance 2.4. What is the number of trials n?
a) 12
b) 10
c) 6
d) 8
Q5. The number of requests reaching a server in one second follows a Poisson distribution with mean 2. What is the probability that at least one request arrives in a given second?
a) 1 − e⁻²
b) e⁻²
c) 1 − 3e⁻²
d) 2e⁻²
Q6. A fair six-sided die is rolled repeatedly until the first six appears. What is the expected number of rolls, including the roll that shows the six?
a) 5
b) 3.5
c) 6
d) 36
Q7. The lifetime of a component is exponentially distributed with mean 10 hours. Given that it has already worked for 5 hours, what is the probability that it works for at least 10 more hours?
a) e⁻¹·⁵
b) 1/2
c) e⁻¹
d) e⁻⁰·⁵
Q8. An exponential random variable has mean 4. What is its median?
a) 4
b) ln 2 / 4
c) 2
d) 4 ln 2
Q9. X is normally distributed with mean 100 and variance 16. Using Φ(2) ≈ 0.9772, what is P(X ≤ 92) approximately?
a) 0.9772
b) 0.1587
c) 0.0228
d) 0.3085
Q10. A fair six-sided die is rolled once. What is the expected value of the outcome given that the outcome is even?
a) 4
b) 3
c) 3.5
d) 5
Q11. A fair coin is tossed. On heads one fair die is rolled; on tails two fair dice are rolled. What is the expected total shown by the dice?
a) 5.25
b) 7
c) 3.5
d) 4.5
Q12. Random variables X and Y have E[X] = 2, E[Y] = 3, E[XY] = 10, Var(X) = 4 and Var(Y) = 9. What is the correlation coefficient of X and Y?
a) 10/6
b) 4/13
c) 4/36
d) 2/3
Q13. For the data 3, 7, 7, 2, 9, 4, 7, 1, which statement about the mean, median and mode is correct?
a) Mean 5.5, median 5, mode 7
b) Mean 5, median 7, mode 7
c) Mean 5, median 4, mode 7
d) Mean 5, median 5.5, mode 7
Q14. A two-state Markov chain moves from state A to state B with probability 0.3 and from B to A with probability 0.2 at each step; otherwise it stays where it is. What is the long-run probability of being in state A?
a) 0.3
b) 0.6
c) 0.4
d) 0.5
Q15. A two-state Markov chain moves from state A to state B with probability 0.3 and from B to A with probability 0.2 at each step; otherwise it stays where it is. Starting in A, what is the probability of being in A after exactly two steps?
a) 0.06
b) 0.55
c) 0.4
d) 0.49
Quick revision notes
- E[aX + b] = aE[X] + b; Var(aX + b) = a²Var(X); for independent X, Y: Var(X ± Y) = Var(X) + Var(Y).
- Binomial: mean np, variance npq; Poisson: P(0) = e⁻λ; geometric trials to first success: mean 1/p.
- Exponential: memoryless, mean 1/λ, median ln 2/λ; normal: Z = (X − μ)/σ.
- Cov = E[XY] − E[X]E[Y]; ρ = Cov/(σₓσᵧ); law of total expectation for two-stage problems.
- Two-state chain stationary: π_A = p(B→A)/(p(A→B) + p(B→A)).
Frequently asked questions
Are these GATE previous year questions?
No, they are original GATE-style practice questions written to the GATE CSE Engineering Mathematics syllabus.
Which distributions are most important for GATE CSE?
Binomial, Poisson, geometric, uniform, exponential and normal. Know their mean and variance, and the memoryless property of the geometric and exponential distributions.
How should I approach two-stage probability problems?
Condition on the first stage, find the expectation or probability in each case, then combine them with the law of total probability or total expectation.
Comments
Post a Comment