ata Y are independent random variables with density functions Jx (x) = e* . u (x) and fy (y) = 2 - e-2y u (y), find the density function of X +X= Z
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- a) Find the joint density function of X and Y. b) Using the method of transformation, find the joint density function of random variables U and V where U = X^2 Y and V = Y .A random variable Y has the density function a) Find E [e^(3Y/2)].b) Find the moment-generating function for Y .c) Use the moment-generating function of Y to find E (Y) and V (Y).The joint density of X and Y is given by f(x,y) = e-(x+y) for 0<x<∞,0<y<∞ and 0 otherwise Find the density function of the random variable Z=X/Y.
- The joint density function of the continuous variables X and Y is fX,Y(x,y) = 7.5x(2-x-y) for 0<X<1 and 0<Y<1. (a) Find the expected value of X+Y. (b) Find fX(x), and fY(y). (c) Find Cov(X,Y). (d) Find Corr(X,Y).Consider the function Find the relationship between the positive constants a and k such that f is a joint density function of the continuous random variables x and yThe joint probability mass function of the two random variables (X, Y) is given by f(x,y) = { 1/5(3x-y), 1<=x<=2 , 1<=y<=3 0 , otherwise• Find the P( X ≤ Y)• Find the marginal density functions of X and Y• Are X and Y independent?• Find E(XY)
- Let X and Y are independent normal random variables with mean 0 and variances 1 and 4 respectively. (a) What is P(Y > X + 3)? (b) What is the joint density function of X and Y ? (c) If (R, Θ) are the polar coordinates of (X, Y ) (that is, R = √ X2 + Y 2 and Θ is the solution to R cos Θ = X and R sin Θ = Y ) then what is the joint density function of R and Θ? (d) Calculate the marginal density function of Θ. please answer all partsA random variable Y has a probability density function f(y) = A(y − 1.5) over the state space−2 ≤ y ≤ 3, where A is a constant. Calculate the median of the random variable Y.dtermine thevalue of c that makes the function f(x,y)=c(x+y) a joint probability density function over the range 0<x<3 and 0<y<3
- A continous random variable X has the density function. f(x) = {x, 0 < x ≤ 1, {2 − x, 1 < x ≤ 2, {0, elsewhere. (a) Show that P(0 < X < 2) = 1 (b) Find P(X < 1.2)ELT(Let) the joint probabilty density function of two con random variable Y and X Find P(Y>=X) Find P(x+y<4)Suppose X,Y have joint density function f(x,y) = (12/7)(xy+y^2), 0 <= x <= 1 and 0 <= y <= 1 and 0, otherwise. a) Check that f is a genuine joint density function. b) Find the marginal density functions of X and Y. c) Calculate the probability P(X < Y).