# Probability And Statistics | Week 9

## Probability And Statistics Week 9 Answers

Q1. Let X₁,…, Xm be a random sample from Poisson distribution with parameters λ, 0 < λ. Which of the following is an unbiased estimator of λ?
(A) X̅, S²
(B) X̅, S², αX̅ + (1 – α)S²; 0 ≤ a ≤1
(C) X̅
(D) S²

Q2. Let three random samples of sizes n₁ = 20, n₂ = 10 and n3 = 8 be taken from a population with mean μ and variance σ2. Then which of the following is an unbiased estimator of σ².
(A) S²1
(B) S²2
(C) S²3
(D) 20S²1+10S²2+8S²3/38

Q3. Let X₁,…, Xn be a random sample from a gamma population with parameters r and λ. Find the moment estimators of λ and r.
(A) X̅, X̅2
(B) 1/X̅, 1/X̅2
(C) X̅/(1/n)Σni=1X2i-X̅2, X̅2/(1/n)Σni=1X2i-X̅2
(D) n-1/X̅, n-1/X̅2

Q4. Let X₁,…, Xn be a random sample from N(0, 1), where θ > 0. Find the MLE of θ.
(A) X̅
(B) 1/X̅
(C) X̅²
(D) 2X̅

Q5. Let X₁,…, Xn be a random sample from exponential distribution with mean μ. Find the MSE of an estimator T = 1/n+1Σni=Xi of μ.
(A) µ²
(B) μ²/n+1
(C) μ²/n(n+1)
(D) nμ²/n+1

Q6. Let X be a Bernoulli random variable with P(X = 1) = p and P(X = 0) = 1-p, 0 < p < 1. If μn denotes the nth moment about mean and μ2n+1 = 0 iff
(A) p = 1/4
(B) p = 1/3
(C) p = 1/2
(D) p = 2/3

Q7. Let X₁,…, Xn be iid random variables with EXi = μ and E|Xi|2 < ∞. Which among the following is a consistent estimator for μ?
(A) (Πni=1/Xi/n
(B) 2X̅
(C) X(1)
(D) 2[n(n+1)]-1Σni=1iXi

Q8. Let -2,-6,5,9,-5,-9 be the observed values of a random sample of size 6 from population having density function given by
fθ(x) = e-(x-θ), x > θ
Then MLE of θ is

(A) -9
(B) 9
(C) 4/3
(D) -4/3

Q9.

(A) (Σxi)/n
(B) (Σxi)/n-1
(C) (Σxi)/2n
(D (2Σxi)/n

Q10. Let X₁,……. Xn be a random sample from U(1,θ) population, where θ > 1. If X(n) = max (X₁, …, Xn), then which of the following is an unbiased estimator of θ ?
(A) n+1/n X(n) +1/n
(B) n+1/n X(n) -2/n
(C) 2X̅ – 1
(D) n/n+1 X(n) + 1/n 0 