5. Let X and Y be independent and identically distributed with marginal probability density function İf a> 0, otherwise, e-ea f(a)-( where >0 (a) [6 pts] Use the convolution formula to fin...
Let X and Y be independent and identically distributed with marginal probability density function f(a)- 0 otherwise, where 8>0 (a) [6 pts] Use the convolution formula to find the probability density function of X +Y. (b) [6 pts) Find the joint probability density function of U X+Y and V- X+Y
9 Let X and Y have the joint probability density function f(x, y) ={4x for 。< otherwise a) What is the marginal density function of Y, where nonzero? b)Are X and Y stochastically independent 9 Let X and Y have the joint probability density function f(x, y) ={4x for 。
2. Let the joint probability density function of (X, Y) be given by {ay otherwise. 1 and 0 < y < 2, f(z,y) (a) [6 pts] Determine if X and Y are independent. (b) [6 pts] Find P{X+Y <1) B( (c) [6 pts) Find 2. Let the joint probability density function of (X, Y) be given by {ay otherwise. 1 and 0
5. If X and Y are independent and identically distributed with Exponential(A), compute El and 6. Let R be the region bounded by the points (0, 1), (-1,0) and (1,0). Joint pdf of (x, Y) is: 1, if (r,y) e R 0, otherwise. Compute P(X-1, γ 7. If X U(0,1) and Y U(0, 1) independent random variables, find the joint pdf of (X+y,x -Y). Also compute marginal pdf of X+Y 8. If x Ezpomential(0.5) and Y ~ Erponential0.5) independent random...
2. Let X and Y be continuous random variables with joint probability density function fx,y(x,y) 0, otherwise (a) Compute the value of k that will make f(x, y) a legitimate joint probability density function. Use f(x.y) with that value of k as the joint probability density function of X, Y in parts (b),(c).(d),(e (b) Find the probability density functions of X and Y. (c) Find the expected values of X, Y and XY (d) Compute the covariance Cov(X,Y) of X...
Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0 otherwise Show that the joint density function of U = 3(X-Y) and V = Y is otherwise, where A is a region of the (u, v) plane to be determined. Deduce that U has the bilateral exponential distribution with density function fu (11) te-lul foru R. Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0...
1. (20 pts) RVs X and Y have joint density function 22 f(x, y) =(0 if O <z<1 and 0<y<2 īf 0 < x < 1 and 0 < y < 2 otherwise (a) Find E(X), V(X), E(Y), and V(Y). (b) Find the covariance cov(X,Y) and the associated correlation ρ (c) Find the marginal densities fx and fy. (Be sure to say where they're nonzero.) (d) Find E(X | Y = 1.5). (e) Are X and Y independent? Give two...
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).
5. Let X and Y have joint probability density function of the form Skxy if 0 < x +y < 1, x > 0 and y > 0, f(x,y)(, y) = { 0 otherwise. (a) What is the value of k? (b) Giving your reasons, state whether X and Y are dependent or independent. (c) Find the marginal probability density functions of X and Y. (d) Calculate E(X) and E(Y). (e) Calculate Cov(X,Y). (f) Find the conditional probability density function...
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...