a)
The PDF of Y is given by:
Now,
So,
b)
The PDF of Z is given by:
Now,
So,
2.6.9 Let X have density function fx(x) = x/4 for 0 < x < 2, otherwise...
2.9.10 Suppose X has density fX(x) = x3/4 for 0 < x < 2, otherwise fx(x) = 0, and Y has density fr (y)-5y4/32 for 0 < y < 2, otherwise fr (y)-0. Assume X and Y are independent, and let Z = X + Y (a) Compute the joint density fx.r(x. y) for all x, y e R (b) Compute the density fz(z) for 2.
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?
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...
2. Let X and Y have joint density f(x.v) = \ şcy? if 0 <x< 1 and 1 <y<2, otherwise. (a) Compute the marginal probability density function of Y. If it's equal to 0 outside of some range, be sure to make this clear. (b) Set up but do not compute an integral to find P(Y < 2X).
2. Let X have probability density function JX2) = 1/2 0<x< 1 3 < x < 4 otherwise Find the cumulative distribution function of X.
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?
8), Let X and Y be continuous random variables with joint density function f(x,y)-4xy for 0 < x < y < 1 Otherwise What is the joint density of U and V Y
(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)
4. Let X and Y have joint probability density function f(x,y) = 139264 oray3 if 0 < x, y < 4 and y> 4-1, otherwise. (a) Set up but do not compute an integral to find E(XY). (b) Let fx() be the marginal probability density function of X. Set up but do not compute an integral to find fx(x) when I <r54. (c) Set up but do not compute an integral to find P(Y > X).
(10 points) Let Y have probability density function (pdf) 3y?, for ( <y<1 fy(y) = 10, otherwise (a) Compute the probability density function (pdf) of 1/Y. (b) Compute the probability density function (pdf) of Y1 +Y2, if Yį and Y2 are inde- pendent random variables with the same pdf as Y. (You can use a computer to help with the integration).