# Probability And Statistics | Week 9

**Course Name: Probability And Statistics**

**Course Link: Click Here**

**These are Probability And Statistics Week 9 Assignment 9 Answers**

**Q1. Let X₁,…, X**_{m} 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, **α**X̅ + (1 – **α**)S²; 0 ≤ a ≤1

(C) X̅

(D) S²

_{m}be a random sample from Poisson distribution with parameters λ, 0 <

**λ**. Which of the following is an unbiased estimator of λ?

**Answer: (B) X̅, S², αX̅ + (1 – α)S²; 0 ≤ a ≤1**

**Q2. Let three random samples of sizes n₁ = 20, n₂ = 10 and n _{3} = 8 be taken from a population with mean μ and variance σ2. Then which of the following is an unbiased estimator of σ².**

(A) S²

_{1}

(B) S²

_{2}

(C) S²

_{3}

(D) 20S²

_{1}+10S²

_{2}+8S²

_{3}/38

**Answer: (D) 20S² _{1}+10S²_{2}+8S²_{3}/38**

**These are Probability And Statistics Week 9 Assignment 9 Answers**

**Q3. Let X₁,…, X _{n} be a random sample from a gamma population with parameters r and λ. Find the moment estimators of λ and r.**

(A) X̅, X̅

^{2}

(B) 1/X̅, 1/X̅

^{2}

(C) X̅/(1/n)Σ

^{n}

_{i=1}X

^{2}

_{i}-X̅

^{2}, X̅

^{2}/(1/n)Σ

^{n}

_{i=1}X

^{2}

_{i}-X̅

^{2}

(D) n-1/X̅, n-1/X̅

^{2}

**Answer: (C) X̅/(1/n)Σ ^{n}_{i=1}X^{2}_{i}-X̅^{2}, X̅^{2}/(1/n)Σ^{n}_{i=1}X^{2}_{i}-X̅^{2}**

**Q4. Let X₁,…, X _{n} 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

**Answer: (A) X̅**

**These are Probability And Statistics Week 9 Assignment 9 Answers**

**Q5. Let X₁,…, X _{n} be a random sample from exponential distribution with mean μ. Find the MSE of an estimator T = 1/n+1Σ^{n}_{i}=X_{i} of μ.**

(A) µ²

(B) μ²/n+1

(C) μ²/n(n+1)

(D) nμ²/n+1

**Answer: (B) μ²/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 μ _{n} denotes the nth moment about mean and μ_{2n+1} = 0 iff**

(A) p = 1/4

(B) p = 1/3

(C) p = 1/2

(D) p = 2/3

**Answer: (C) p = 1/2**

**These are Probability And Statistics Week 9 Assignment 9 Answers**

**Q7. Let X₁,…, X _{n} be iid random variables with EX_{i} = μ and E|X_{i}|^{2} < ∞. Which among the following is a consistent estimator for μ?**

(A) (Π

^{n}

_{i=1}/X

_{i})¹

^{/n}

(B) 2X̅

(C) X

_{(1)}

(D) 2[n(n+1)]

^{-1}Σ

^{n}

_{i=1}iX

_{i}

**Answer: (D) 2[n(n+1)] ^{-1}Σ^{n}_{i=1}iX_{i}**

**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 byf _{θ}(x) = e^{-(x-θ)}, x > θThen MLE of θ is**

(A) -9

(B) 9

(C) 4/3

(D) -4/3

**Answer: (A) -9**

**These are Probability And Statistics Week 9 Assignment 9 Answers**

**Q9.**

**Answer: (C)**

**Q10. Let X₁,……. Xn be a random sample from U(1,θ) population, where θ > 1. If X _{(n)} = max (X₁, …, X_{n}), 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

**Answer: (C) 2X̅ – 1**

**These are Probability And Statistics Week 9 Assignment 9 Answers**

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**These are Probability And Statistics Week 9 Assignment 9 Answers**

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