# An Exponential Distribution

1. A statistician has a sample X1, . . . , Xn from N(µ, ? 2). Also, the statis-tician has a sample Y1, . . . , Yk from an exponential distribution with param- eter ?. Suppose that these samples are independent. Find the variance of a random variable Z = 3X? ? 5Y? .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 E?(Xl) = l?. Consider an es- timator ?? =?n l=1 alXl. What condition should be imposed on a1, a2, . . . , anso that ?? is an unbiased estimator? 4. Let X1, . . . , Xn be iid Bernoulli with the probability of success ?. 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(?, ?). Find (minimal) sufficient statistic for the pair (?, ?).6. Consider a sample of size n from Unif(0, ?). Find a method of mo- ments estimator of ?.7. Let we observe a sample X1, . . . , Xn where Xl = ? + Yl, with Yl being iid Expon(?). Find the MLE of ?. Hint: Do not forget about support of the exponential RV.8. For the problem 7, let ? be given. Find the MLE of cos(?). 9. Consider a sample of size n from Poisson(?). Find a method ofmoments estimator for the estimand ?2. 10. Let a sample of size n from N(?, ?2) be given. A statistician believes

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