You may use the following facts to answer the questions below Fact 1: Suppose that Xi. . . . , X, are independent and X.* GAM (θ.k.) for -1 -1 Fact 2: If Y GAM(0,n aYGAM(ab,n) for any number a &g...
Let xi be independent. E(xi)=0. Var(xi)= sigma ^2 Cov(x,y) = E(XY) - ExEy Use this fact and apply it to this example ! Do not use anything that has not been giving. I’m having difficulties completing this problem. Check pictures to see how I done a far AaBbCcDdEe AaBbCc Normal No Spac hoose Check for Updates. に1 に2 We were unable to transcribe this image
50] 1. Suppose that Xi,X2.. are independent and identically distributed Bernoulli random vari-ables with success probability equal to an unknown parameter p E (0, 1). Let P,-n-1 Σǐl Xi denote the sample proportion. liol a. Ti, what des VatRtA-P) converge in law ? 10 a. To what does)converge in law ? [10] b. Use your answer to part a to propose an approximate 95% confidence interval for p. 10 c. Find a real-valued function g such that vn(g(p) -g(p)) converges...
Suppose that Xi, X2,..., Xn are independent random variables (not iid) with densities x, (x^, where 6, > 0, for i-1, 2, , n. versus H1: not Ho (c) Suppose Ho is true so that the common distribution of X1, X2,..., Xn, now viewed as being conditional on 6, is described by where θ > 0. Identify a conjugate prior for 0. Specify any hyperparameters in your prior (pick values for fun if you want). Show how to carry out...
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
2. Suppose you decide to randomly generate numbers from X ~ Unif(0, ). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use alLE = max Xi = X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead, he'll say the second largest number: θ = Xn-1. Determine the bias of this estimator...
Suppose X, Y are independent with X ∼ N (0, 1) and Y ∼ N (0, 1). Show that the distribution of Q = X/Y follows the Cauchy distribution, i.e., f(q) = 1/π(1+q2) . Hint: Let Q = X/Y and V=Y. Find the joint pdf of Q and V and finally find the marginal pdf of Q by integrating the joint pdf of Q and V w.r.t. V: Y π(1+q2) Y V = Y . Find the joint pdf of...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo 5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
1. Suppose Z N(0, 1) ει ~ N(0, ơÐ €2 ~ N(0,03) independent and let (a) 12 pts] Under what conditions (if an (b) [2 pts] Determine the covariance of Yi and Y2. Under what conditions (if any) are they y) are Yǐ and Y2 exchangeable? Justify your answer. (marginally) independent? 1. Suppose Z N(0, 1) ει ~ N(0, ơÐ €2 ~ N(0,03) independent and let (a) 12 pts] Under what conditions (if an (b) [2 pts] Determine the covariance...
explan the answer 7. Let X~ N(0, 1) and let Y xi. Find the probability density function of Y and, hence or otherwise, find the mean and variance of Y .
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...