Let f(x, y) be the joint probability density function of X and Y defined by x) = {².- Find Cov(X, Y). f(x,y) (2x +2y - 4xy, if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1; otherwise.
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- For the probability density function f(x) = 3x^2 on [0,1], find: V(X)Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independentSuppose that the random variables X and Y have a joint density function f(x,y).prove that Cov(X,Y)=0 if E(X|Y=y) does not depend on y
- Suppose that X and Y have a joint probability density function f(x,y)= 1, if0<y<1,y<x<2y; 0, otherwise. (a) Compute P(X + Y less than or equal 1). (b) Find the marginal probability density functions for X and Y , respectively. (c) Are X and Y independent?Suppose the joint probability density of X and Y is fX,Y (x, y) = 3y 2 with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and zero everywhere else. 1. Compute E[X|Y = y]. 2. Compute E[X3 + X|X < .5]If X is uniformly distributed over (0,1), find the density function of Y = eX
- Let X be a continuous random variable with density function f(x) = 2x, 0 ≤ x ≤ 1. Find the moment-generating function of X, M(t), and verify that E(X) = M′(0) and that E(X2) = M′′(0).Calculate the E(X) when the joint probability density function of X and Y is fxy(X,Y)=c(X+Y) over the range x = 1, ..., 4 and y = 1, ..., 2Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise Find the constant c, P(Y≥1/2), P(X < 2, Y >1/2), P(X < 1), Determine whether X and Y are independent.
- The PDF of a continuous random variable X is as follows: f(X)= c(4x2 - 2x2) 0<* x <* 2 (*less or equal to) a. For this to be a proper density function, what must be the value of c ?Find a value of k that will make f a probability density functionon the indicated interval.ƒ(x) = kx; [0, 3]Suppose X and Y have a joint density function of f(x,y)=6x2y, 0<x<1, 0<y<1. Find P(X+2Y<1).