Midterm 1, STAT 3445 — Introduction to Mathematical Statistics II,Spring 2021The exam is designed to be finished in 3 hours. There are 50 points in total. The exam should be done individually and similar solutions will be penalized with a final grade of ”zero”. The problems are not necessarily in the order of difficulty level. No credit will be given to results without necessary justification. good luck!

1. (10 points) LetY1,Y2,…,Yn be independent and identically distributed random variables such that for0< p <1,P(Yi= 1) = 1−p and P(Yi= 0) =p.(a) (2 points) Find moment generating function for the random variableY1(b) (2 points) Find the moment generating function for W=Y1+Y2+…+Yn(c) (3 points)T1= 1− ̄Yis an unbiased estimator of p? Find the MSE ofT1(d) (3 points) Construct an approximated two-sided (1−α) confidence interval for p.

2. (10 points) LetX1,X2,…,Xn be i.i.d. Exp(θ) whereθ >0 is unknown.(a) (2 points) Show that ̄Xis an unbiased estimator forθ(b) (2 points) Show that nX(1)= n×min(X1,X2,…,Xn) is an unbiased estimator forθ(c) (3 points) Based on the the MSE of each estimator, which estimator is better for estimatingθ, ̄XornX(1)?(d) (3 points) UsingX1,X2,…,Xn, (i.e. using all available Random Variables) construct an unbiased estimator for1θ(you may assume that n >1)

3. (10 points) LetX1,X2,…,X n be i.i.d. Geo(p). For each Xi, with the probability mass function P(x) =p(1−p)x, where x= 0,1,2,…(a) (2 points) Derive the distribution of U=∑ni=1Xi(b) (3 points) Find an unbiased estimator,W, of 1p, that is a function ofX1,X2,…,Xn. Confirm that your estimator is unbiased.(c) (3 points) Find MSE(W), the unbiased estimator that you found in the previous part.(d) (2 points) Suppose that the value of p is known. Create a statistic Y that is both a function ofX1,X2,…,X nand is approximately distributed N(0,1) when n is sufficiently large.

4. (10 points) Suppose you have a sample of sixteen independent observationsY1,Y2,…,Y16from a normal population with mean 1 and variance 5. ̄YandS2Yare the sample mean and sample variance, respectively.What is the distribution of(a) (2 points) 16( ̄Y−1)2/5.(b) (2 points)S2Y.(c) (3 points) 4( ̄Y−1)/SY.(d) (3 points) Suppose thatX1,…,X4are from an independent normal population with mean 0 and variance 1. DefineS2X=134∑i=1(Xi− ̄X)2. What is the distribution of 3(S2X+S2Y)?

5. (10 points) The feeding habits of two species of net-casting spiders are studied. The species, the deinopis and menneus, coexist in eastern Australia. The following summaries were obtained on the size, in millimeters, of the prey of random samples of the two species:Deinopis:n1= 50 ̄x1= 10.5, s1= 2.5Menneus:n2= 50, ̄x2= 9.5, s2= 1.9(a) (2 points) Find a point estimate forμ1−μ2, the difference in average of the size of the prey.(b) (2 points) State your assumptions to construct an exact (1−α) confidence interval forμ1−μ2.(c) (5 points) Construct a two-sided 95% confidence interval forμ1−μ2.(d) (1 points) Draw a conclusion