Problem 5 The joint pdf of x1 and x2 is 0.2 0 0.2 X1=1 | 0.2 0.2 X1=3 | 0.2 a) Find marginal pdfs p1(x1) and p2(x2) b) Are x1 and x2 independent? c) Compute Efx1 + x2J d) Compute covfx,, x2) e) Compute vars5x1 - 6x2}
(a) Write down the joint pdf of X1 and X2. [4] (b) By using the transformation of random variable method, find the joint pdf of Y1 = X1 and Y2 = X2/X1. [16] (c) Hence find the marginal pdfs of Y1 and Y2. [8] (d) Compute the covariance between Y1 and Y2, cov [Y1, Y2]. [8] (e) State, with justification, whether Y1 and Y2 are independent.
Suppose X1 and X2 are continuous random variables with X1 ~ Unif(0, 1), X2 | X1 = x1 ~ Unif(0, X1) (a) Find the pdf for the joint distribution of X1 and X2 (b) Find the pdf for the marginal distribution of X1 (c) Find the pdf for the marginal distribution of X2 (d) Find the pdf for the conditional distribution of X1 | X2 = x2 (e) Write 1 or 2 sentences explaining how this problem relates to Bayes’...
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.
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
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.)
Let X1 and X2 have joint PDF f(x1,x2)=x1+x2 for 0 <x1 <1 and 0<x2 <1.(a) Find the covariance and correlation of X1 and X2. (b) Find the conditional mean and conditional variance of X1 given X2 = x2.
1. Let X1, X2, X3 be continuous random variables with joint probability density function 00 < Xi < 00,i=1,2,3 Consider the transformation U-X1, V = X , W-XY + X + X (a) Find the joint pdf (probability density function) of U, V and W. (b) Find the marginal pdf of U, and hence find E(U) and Var(U) (c) Find the marginal pdf of W, and hence find E(W) and Var(W) (d) Find the conditional pdf of U given Ww,...
2. Let the pair (X,Y) have joint PDF fxy(x, y) = c, with 2.2 + y2 <1. (a) Find c and the marginal PDFs of X and Y. (b) What are the means of X and Y ? No calculations are needed, only a brief expla- nation is required. (c) Find the conditional PDF of Y given X = x and deduce E|Y|X = x]. (d) Obtain E(XY) and compare it to E[X]E[Y). (e) Are X and Y independent? Explain....
Suppose that (X1, X2) N (0,0,1,1,0). It follows from this that the joint PDF of (X1, X2) is given by Ixvx:(21,2) = exp (1} (27 +23)) Furthermore, if 1 Y Tā (X1 + X2) and Y2 (X1 - X2) Then (Y1,Y) ~ N(0,0,1,1,0) as well. (a) If X1 <X2, what are the possible values of Y¡ and Y2? (b) If Y, <0, what are the possible values of Xi and X,? (c) What is the marginal distribution of Yg? (d)...