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.
1. (10 points) LetY1,Y2,…,Ynbe independent and identically distributed random variables such that for0< p <1,P(Yi= 1) = 1−pandP(Yi= 0) =p.(a) (2 points) Find moment generating function for the random variableY1(b) (2 points) Find the moment generating function forW=Y1+Y2+…+Yn(c) (3 points)T1= 1− ̄Yis an unbiased estimator ofp? Find the MSE ofT1(d) (3 points) Construct an approximated two-sided (1−α) confidence interval forp.
2. (10 points) LetX1,X2,…,Xnbe 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 unbiasedestimator for1θ(you may assume thatn >1)
3. (10 points) LetX1,X2,…,Xnbe i.i.d. Geo(p). For eachXi, with the probability mass functionP(x) =p(1−p)x, wherex= 0,1,2,…
(a) (2 points) Derive the distribution ofU=∑ni=1Xi
(b) (3 points) Find an unbiasted estimator,W, of1p, that is a function ofX1,X2,…,Xn. Confirm thatyour 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,…,Xnand 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 normalpopulation 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 andvariance 1. DefineS2X=134∑i=1(Xi− ̄X)2. What is the distribution of 3(S2X+S2Y)?1
5. (10 points) The feeding habits of two species of net-casting spiders are studied. The species, the deinopisand menneus, coexist in eastern Australia. The following summaries were obtained on the size, in mil-limeters, 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