2.9.10 Suppose X has density fX(x) = x3/4 for 0 < x < 2, otherwise fx(x)...
2.6.9 Let X have density function fx(x) = x/4 for 0 < x < 2, otherwise fx(x)=0. (a) Let Y = X. Compute the density function fy(y) for Y. (b) Let Z = X. Compute the density function fz(z) for Z.
Suppose X and Y are two continuous random variables with probability density functions: fx(x)1 for 1<x2, fx(x) 0 otherwise, and fr (v) 3e3y for y>0, fr (y) 0 otherwise. a) Suppose X and Y are independent, is Z-X+ Y"memoryless"? Justify your answer. b) Suppose that the conditional expected value satisfies E(Y X)-X. Find Cov0), and El(Y-X) expX)]. Suppose X and Y are two continuous random variables with probability density functions: fx(x)1 for 10, fr (y) 0 otherwise. a) Suppose X...
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).
Problem 5. The joint density of X and Y is given by e" (z+y) fx.-otherwise. İf 0 < x < oo, 0 < y < 00, Consider the random variable Z-; a) Find the cumulative distribution function of Z b) What is the probability density function 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?
(4) Suppose that the joint density function of X, Y and Z is given by )<y <<< 1 f(x, y, z) = { otherwise. (a) Find the marginal density fz(z) (b) Find the marginalized density fxy(x, y) 72 (c) Find E (2)
fx (z)='0 otherwise Let Xa)<...<Xn) be the order statistics. Show that Xa)/X(n) and X(n) are independent random variables.
Do only (b). Use convolution. Exercise 7.20. Let X have density fx(x) = 2x for 0 < x < 1 and let Y be uniform on the interval (1,2). Assume X and Y independent. Give the joint density function of (X. Y). Calculate P(Y - X2 Find the density function of X + Y. (a) (b)
c 3. Let X have density fx () = 1+1 -1<<1. (a) Compute P(-2 < X <1/2). (b) Find the cumulative distribution Fy(y) and probability density function fy(y) of Y = X? (c) Find probability density function fz() of Z = X1/3 (a) Find the mean and variance of X. (e) Calculate the expected value of Z by (i) evaluating S (x)/x(x)dr for an appropriate function (). (ii) evaluating fz(z)dz, pansion of 1/3 (ii) approximation using an appropriate formula based...
Consider fx (x)=e*, 0<x and joint probability density function fx (x, y) = e) for 0<x<y. Determine the following: (a) Conditional probability distribution of Y given X =1. (b) ECY X = 1) = (c) P(Y <2 X = 1) = (d) Conditional probability distribution of X given Y = 4.