Let (X, Y ) be a random point in the square {(x, y)| 0 ≤ x, y ≤ 1}. Compute the density of W = XY , E[W] and Var(W)
Let X ~ N(0, 1) and let Y be a random variable such that E[Y|X=x] = ax +b and Var[Y|X =x] = 1 a) compute E[Y] b) compute Var[Y] c) Find E[XY]
3. Let the random variables X and Y have the joint probability density function 0 y 1, 0 x < y fxy(x, y)y otherwise (a) Compute the joint expectation E(XY) (b) Compute the marginal expectations E(X) and E (Y) (c) Compute the covariance Cov(X, Y)
Please answer everything and give a detailed answer. Thanks 2. Let (X, Y) be a continuous random vector with probability density function 2xety, if x 2 0 and 1 < y< 0, 2, else. (c) Find the moment generating function of X; using the moment gener-ating function, calculate Var(X2) (d) Calculate Cov(X, Y). Calculate Var(X +Y) and Var(X -Y). Calculate P(XY 2 2XY 2 1) 2. Let (X, Y) be a continuous random vector with probability density function 2xety, if...
Let the random variable X and Y have the joint probability density function. fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
3. Let the random variables X and Y have the joint probability density function fxr (x, y) = 0 <y<1, 0<xsy otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
(II) Multiple continuous random variables: 8.2 Let X and Y have joint density fXY(x,y) = cx^2y for x and y in the triangle defined by 0 < x < 1, 0 < y < 1, 0 < x + y < 1 and fXY(x,y) = 0 elsewhere. a. What is c? b. What are the marginals fX(x) and fY(y)? c. What are E[X], E[Y], Var[X] and Var[Y]? d. What is E[XY]? Are X and Y independent?
Let X and Y be two independent random variables with X =d R(0, 2) and Y =d exp(1). (a) Use the convolution formula to calculate the probability density function of W =X+Y. (b) Derive the probability density function of U = XY .
The joint density of random variables X and Y is given to be f(x,y) =xy^2 for 0≤x≤y≤1 and is 0 elsewhere. (a) Compute the marginal densities for X and for Y respectively. (b) Compute the expected valueE(XY). (c) Define a new random variable W=Y/X. Compute the probability P(W > t) for anyt >1. Also find the probability P(W <1/2) ?
Choose a point at random in the square with sides 0 <=x≤1 and ≤ y ≤ 1. This means that the probability that the point falls in any region within the square is the area of that region. Let X be the x coordinate and Y be the y coordinate of the point chosen. Find the conditional probability Pr(Y<1/3|Y>X). Hint Sketch the square and the events Y<1/3 and Y>X
Let X be an exponential random variable with parameter A > 0, and let Y be a discrete random variable that takes the values 1 and -1 according to the result of a toss of a fair coin Compute the CDF and the PDF of Z = XY Let X be an exponential random variable with parameter A > 0, and let Y be a discrete random variable that takes the values 1 and -1 according to the result of...