3 lat (0.142.0 person eye sependent at the mom 8. Let {N(t),t > 0} be a...
For a random walk with random starting value, let Y, Yoterter-1e for t > 0, where Yo has a distribution with mean μ0 and variance σό . Suppose fur- ther that Yo, et.., e are independent. (a) Show that E(Y) Ho for all t. (b) Show that Var(,) = tơ24 (c) Show that Cody, Y.) = min(t, s) + 05, , lienee , that cov ( var (it) and (d) Show that Corr(,) = 1 for 0st s s.
is independent of X, and e Problem 3 Suppose X N(0, 1 -2) -1 <p< 1. (1) Explain that the conditional distribution [Y|X = x] ~N(px, 1 - p2) (2) Calculate the joint density f(x, y) (3) Calculate E(Y) and Var(Y) (4) Calculate Cov(X, Y) N(0, 1), and Y = pX + €, where
Let X, Y be random variables with f(x, y) = 1,-y < x < y, 0 < y < 1. Show that Cov(X,Y) = 0. Are X, Y independent?
1. Let (N(t))>o be a Poisson process with rate X, and let Y1,Y2, ... bei.i.d. random variables. Fur- ther suppose that (N(t))=>0 and (Y)>1 are independent. Define the compound Poisson process N(t) Y. X(t) = Recall that the moment generating function of a random variable X is defined by ºx(u) = E[c"X]. Suppose that oy, (u) < for all u CR (for simplicity). (a) Show that for all u ER, ºx() (u) = exp (Atløy, (u) - 1)). (b) Instead...
3) Suppose X~N(0,1) and Y~N(2,4), they are independent, then is incorrect. 6 X-Y N(-2,5) D Var(X) < Var(Y) SupposeX-N(Aof) and Y-N(H2,σ ), they arc indcpcndcnt, thcn in the following statementss incorrect 4) 5) Suppose X~NCHiof) and Y~NCHz,σ ), they are independent, if PCIX-Hik 1) > PCIY _ μ2I 1), then ( ) is correct.
2. Let Xn, n > 1, be a sequence of independent r.v., and Øn (t) = E (eitX»), ER be their characteristic functions. Let Yn = {k=0 Xk, n > 0, X0 = 0, and 8. () = {1*: (),ER. k = 1 a) Let t be so that I1=1 løk (t)) > 0. Show that _exp{itYn} ?, n > 0, On (t) is a martingale with respect to Fn = (Xo, ...,Xn), n > 0, and sup, E (M,|2)...
Let > 0 and a > 0 be given. Suppose that X is a random variable with moment generating function e My(t) = {(A-ta tsy Top til Compute Var(X). Show that if we define Ly(t) = In My(t) then Ls (0) = Var(X).
QUESTION 14 If (S-1) <0, and (T - G) <0, then (M - X)>0 True False
Help please! Let Be be Brownian motion and fix to > 0. Prove that By: = Bto+t - Blo; t o is a Brownian motion.
have a Ba- Question 2. (20 points) Let (X1,...,xn) be a sample from Poisson(2), and let prior distribution Gamma(a,b), a,ß > 0, with pdf fe exp(-82), when 1 2 0, (A) =1 Ta) * 10, when 1 <0, where I(a):= 6°40-1 exp(-t)dt, for a > 0. (a). (10 points) Find the posterior distribution of 1. (0.1) (b). (5 points) Calculate the posterior mean and variance. (c). (5 points) Now consider a Bayesian test of H: 15 lo versus H:2 >...