statistics

1. A statistician has a sample X1, . . . , Xn from N(µ, σ 2). Also, the statis-

tician has a sample Y1, . . . , Yk from an exponential distribution with param- eter γ. Suppose that these samples are independent. Find the variance of a random variable Z = 3X̄ − 5Ȳ .

2. Let a sample of size n from a distribution with the pdf f(x) be given. What is the pdf of X(1)?

3. Let X1, . . . , Xn be independent RVs with Eθ(Xl) = lθ. Consider an es- timator Θ̂ =

∑n l=1 alXl. What condition should be imposed on a1, a2, . . . , an

so that Θ̂ is an unbiased estimator? 4. Let X1, . . . , Xn be iid Bernoulli with the probability of success θ. Sug-

gest the minimal variance unbiased estimator, and then prove, using Cramer- Rao inequality, its efficiency.

5. Consider a sample of size n from Unif(α, β). Find (minimal) sufficient statistic for the pair (α, β).

6. Consider a sample of size n from Unif(0, θ). Find a method of mo- ments estimator of θ.

7. Let we observe a sample X1, . . . , Xn where Xl = θ + Yl, with Yl being iid Expon(λ). Find the MLE of θ. Hint: Do not forget about support of the exponential RV.

8. For the problem 7, let θ be given. Find the MLE of cos(λ). 9. Consider a sample of size n from Poisson(λ). Find a method of

moments estimator for the estimand λ2. 10. Let a sample of size n from N(θ, σ2) be given. A statistician believes

that θ is a realization of a normal random variable Θ ∼ N(µ, ν2). Find the Bayesian estimator of θ. [Please write down all your steps — just an answer will not be counted.]

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