Question

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 times) a technician is called on an emergency call. The joint probability distribution fxy(x,y) is given by * X 1 / 2 1 0.05 20.05 3 0 0.05 0.1 0.1 0.35 0.1 0.2 (a) Evaluate the marginal pdf and the mean of X. (b) Evaluate the marginal pdf and the mean of Y. (c) Evaluate P(Y = 3 X = 2). (d) Evaluate E (XY) and determine if X and Y are uncorrelated. 3. Random variables X and Y have the joint distribution x<0 or y< 0 5x + e-(x+1) Fxy(x,y) = 4 -32 0<x<4 x e + 1 +1 uz 5 Ley, 45 x and any y > 0 (a) Find the marginal distributions Fx(x) and Fy(y). (b) Compute the joint probability P(3 < X < 5,1 <Y S 2). 0<y<1 fxy(x,y) = 0, 4. The random variables X and Y have joint density (12xy(1-x), 0<x<1, otherwise (a) Find the marginal pdfs of X and Y. (b) Find the mean and variance of X and Y. (c) Find the conditional distributions fxy(x|y) and fylx(y|x). (d) Are X and Y independent?

Answer 1

1) The given joint PDF is .

fx.r(x,y) = A(1 - x)e-V; 0<x<1,0 <y <a

a) The condition for PDF is

fx,y(x,y)dxdy = 1 Ae | (1-X)dxdy = 1 ए|N (1) = 1 |N

b) The marginal PDFs are

fy(y) = (fx,x(x,y)dx fy(Y) = Aev 1 (1 - x)dx fr (Y) = 4ev fy(y) = e-Y;0<y < 0

fx(x) = "fx,x(x,y)dy fx(x) = A ( M (1 - x)dy fx (X) = A(1 - x) /e Ydy JO fx(x) = 2 (1-x); 0<x<1

c) The expectations are

E(X) = 5 *(x)dx E(X) = 1*2*(1-x)dx E(X) = 2 [(x – x2) dx E(X) = 2 (x2/2 – x3/3) E(X) = WENNS

E(Y) = 1 °yfr (y)dy E(Y) = ( ye Ydy E(Y) = 1

d) The expected value

0 1 E(XY) = f ( xyfx,x(x,y)dxdy E(XY) = 26" [xy (1-x) e Ydxdy E(XY) = 2 (yev (x(1 - x) dxdy E(XY) = 3 | ºye Ydy E(XY) = Jo

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