Graybill, 1961]. Let x -(X1, X2) have a bivariate normal distribution with pdf where Q-2x-x1x2 +...
Let X = (X1, X2) be a bivariate normal random vector such that Mi = 4,42 = 6,01 = 25, 02 = 16 and p= 0.7. 1. Find P(X2 <5|X1 = 3).
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
Exercise 7 (team 5) Let Xi and X2 have joint pdf x1 + x2 if0<x1 < 1 and 0 < x2 < 1 /h.x2 (x1,x2) = 0 otherwise. When Y1 X1X2 derive the marginal pdf for Y.
3. Let X ~ N2(u, ) be a bivariate Normal random vector, where 6-61-10 X = = | 12 (a) Find the distribution of Y1 = X1 + X2. (b) Let Y2 = X1 +aX2. Find value a such that Yį and Y2 are independent.
Let X1 and X2 be independent standard normal random vari ables, and let Y-AX-b, where Y-(y, Y)т, X-(X1, X2)T, b (1, -2) and (a) Determine the joint pdf of ı and Y2 by using the formula given in class for the joint pdf of Y = g(X) when X and Y are random vectors of the same dimension, and q is invertible with both g and its inverse differentiable (b) Show that the joint pdf in (a) can be expressed...
Problem 1. (Bivariate Normal Distribution) Let Z1, Z2 be i.i.d. N(0,1) distributed random variables, and p be a constant between –1 and 1. define X1, X2 as: x3 = + VF5223X = v T14:21 - VF52 23 1) Show that, (X1, X2)T follows bivariate Normal distribution, find out the mean vector and the covariance matrix. 2) Write down the moment generating function, and show that when p= 0, X11X2.
5. Let X1, X2, ..., Xn be a random sample from a distribution with pdf of f(x) = (@+1)xº,0<x<1. a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for @ ?
Please do by hand. Thanks in advance. 5. Let X1 and X2 have joint pdf f(x1, x2) = 4xı, for 0 < x < x2 < l; and 0 otherwise. Find the pdf of Y = X/X2. (Hint: First find the joint pdf of Y and Y2 = X1.)
5. Let X1, X2,. , Xn be a random sample from a distribution with pdf of f(x) (0+1)x,0< x<1 a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for 0?
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5 ], = [11] Here -1 <p<1. (a) What is P(X > Y)? (b) Is there a constant c such that X and X +cY are independent?