7. Let X and Y have joint pdf 122 (1-x)y, 0, 0〈x〈1,0くyく1. otherwise. x,y(x,y) = (a) Find the joint cdf of X and Y. (4pts) (b) Find PY< VX. (Spts) (c) Find the marginal pdfs of X and Y. (6pts)...
1) Let X and Y have joint pdf: fxy(x,y) = kx(1 – x)y for 0 < x < 1,0 < y< 1 a) Find k. b) Find the joint cdf of X and Y. c) Find the marginal pdf of X and Y. d) Find P(Y < VX) and P(X<Y). e) Find the correlation E(XY) and the covariance COV(X,Y) of X and Y. f) Determine whether X and Y are independent, orthogonal or uncorrelated.
2. Let the random variables X and Y have the joint PDF given below: (a) Find P(X + Y ≤ 2). (b) Find the marginal PDFs of X and Y. (c) Find the conditional PDF of Y |X = x. (d) Find P(Y < 3|X = 1). Let the random variables X and Y have the joint PDF given below: 2e -0 < y < 00 xY(,) otherwise 0 (a) Find P(XY < 2) (b) Find the marginal PDFs of...
5. Let the joint density of X and Y be fr(x,) = (x + y, fxy(x, y) = 0, 0<x< 1,0 <y <1 otherwise (a) Find the marginal pdfs of X and Y. (b) Are X and Y independent? (c) Are X and Y correlated? (d) Find P(X + Y < 1).
Let X and Y be continuous rvs with a joint pdf of the form: ?k(x+y), if(x,y)∈?0≤y≤x≤1? f(x,y) = 0, otherwise (a) Find k. (b) Find the joint CDF F (x, y). 0, otherwise (c) Find the conditional pdfs f(x|y) and f(y|x) (d) Find P[2Y > X] (e) Find P[Y + 2X > 1]
24. The joint cdf of (X,Y) is Find a) Joint pdf of (X, Y) b) Marginal pdf of X and Y c) PI(X s 1) n (Y s 1) d) PI(1 < X <3) n (1 <Y <2)] Page 4 of5
Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint pdf given by fx,YZ(xgz) = k xyz if 0 S$ 1,0 rS 1,0 25 1 ) and fxyZ(x,y,z) = 0, otherwise. (a) Find k so that fxyz(x.yz) is a genuine probability density function. (b) Are X,Y,Z independent? (c) Find PXs 1/2, Y s 1/3, Z s1/4). (d) Find the marginal pdf fxy(x.y). (e) Find the marginal pdf fx(x). Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint...
Suppose X and Y have the joint pdf f (x, y) = 3y, 0 < y < 1, y − 1 < x < 1 − y 0 otherwise a) Give an expression for P (X > Y ). b) Find the marginal pdfs for Y . c) Find the conditional pdf of X given Y = y, where 0 < y < 1. d) Give an expression for E[XY ]. e) Are X and Y independent?
4. Two RVs with a joint pdf given as follows fx.x ), 0<x< 1,0 <y<1 otherwise (a) Find fr ). (6 point) (b) Find fxy(x[y). (6 points) (c) Are X and Y independent? (clearly show justification for credit) (6 points)
Let X and Y have the joint pdf f(x,y) = e-x-y I(x > 0,y > 0). a. What are the marginal pdfs of X and Y ? Are X and Y independent? Why? b. Please calculate the cumulative distribution functions for X and Y, that is, find F(x) and F(y). c. Let Z = max(X,Y), please compute P(Z ≤ a) = P(max(X,Y) ≤ a) for a > 0. Then compute the pdf of Z.
Let X, Y be jointly continuous with joint density function (pdf) fx,y(x, y) *(1+xy) 05 x <1,0 <2 0 otherwise (a) Find the marginal density functions (pdf) fx and fy. (b) Are X and Y independent? Why or why not?