2. Suppose X - Unif (0, 1) and S, |X ~ Bin(n, X). Let I, indicate the ith trial is a success. This 10, find: implies that llx ~iid Bern(p a) P(S1o 3) X). For n c) P(I11 1S10 3) d) P(l111, 12 1S10...
15. Suppose Ui ~ iid Unif(0, 1) for n = 6. Let X = U(1), Y = U(6), and W = X/Y. Find: ~Ll b) Fw(w) c) E(W) d) Var(W)
1. Let X ~ Bin(n = 12, p = 0.4) and Y Bin(n = 12, p = 0.6), and suppose that X and Y are independent. Answer the following True/False questions. (a) E[X] + E[Y] = 12. (b) Var(X) = Var(Y). (c) P(X<3) + P(Y < 8) = 1. (d) P(X < 6) + P(Y < 6) = 1. (e) Cov(X,Y) = 0.
Let X,, X,,... be independent and identically distributed (iid) with E X]< co. Let So 0, S,X, n 2 1 The process (S., n 0 is called a random walk process. ΣΧ be a random walk and let λ, i > 0, denote the probability 7.13. Let S," that a ladder height equals i-that is, λ,-Pfirst positive value of S" equals i]. (a) Show that if q, then λ¡ satisfies (b) If P(X = j)-%, j =-2,-1, 0, 1, 2,...
2. Suppose that X1, X2, .. , Xn are iid N(0, 02). Where i and o both assumed to be unknown. Let 0 = (i,a). Find jointly sufficient statistics for 0
2. Let X be a binomial variable with n=10. Suppose E(X) - 3. (a) (5 points) What is the probability of success of the binomial experiment that generates the variable X? Explain your answer. (b) (5 points) Find P(X = 3). (c) (5 points) Find P(3 < X < 7). (d) (5 points) Find P(X 1 )
1. Let X~b(x; n, p) (a) For n 6, p .2, find () Prx> 3), (ii) Pr(x23), (ii) Pr(x (b) For n = 15, p= .8, find (i) Pr(X-2), (ii) Pr(X-12), (iii) Pr(X-8). (c) For n 10, find p so that Pr(X 2 8)6778. く2). 2. Let X be a binomial random variable with μ-6 and σ2-2.4. Fin (a) Pr(X> 2) (b) Pr(2 < X < 8). (c) Pr(Xs 8). 1. Let X~b(x; n, p) (a) For n 6, p...
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics. 3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
real analysis proof Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1 Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1
8. Let X.(i-12) be independent N(0,1) random variables. a. Find the value of c such that P ( (X1 + X2尸/( X2-X)2 < c ) =.90 b. Find P(2 X1 -3 X21.5) c. Find 95th percentile of the distribution of Y-2X -3X2
Ques 3 (d) Suppose that n-10, and Xi Xio represent the waiting times that the 10 people must wait at a bus stop for their bus to arrive. Interpret the result of (c) in the context of this scenario be iid observations from the Uniform(0,0) distribution. 3. Again, let X..., X (a) Find the joint pdf of Xu) and X() (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (e) It turns out, if X...