Let X and Y be continuous random variables with joint density function { if x E R, y > 0 f(x, y) = otherwise. Find the joint density function of U = X+Y,V = X - Y.
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- Suppose 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 yThe random variables X and Y have a joint density function f(x, y). f(x,y)= 2x+2y 0<x<1, 0<y<1, y>x 0 elsewhere P(X < 0.5, Y < 0.5) = ______________________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 independent
- Let X, Y be random variables, suppose X is N(0, 1) and Y = e2X. a) Obtain the density function of Y .b) Calculate E(Y) and V ar(Y).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 Find the constant c, P(Y≥1/2), P(X < 2, Y >1/2), P(X < 1), Determine whether X and Y are independent.Let (X, Y ) be a random vector with joint density function of the form:f(X,Y )(x, y) = 24xy, 0 < x < 1, 0 < y < 1, x + y < 10 , otherwise(d) Check that E(E(X | Y )) = E(X).
- For random variables X and Y with joint density function f(x,y) = 6e-2x-3y. (x,y > 0) and f(x,y) = 0, otherwise, find: a) P(X <= x, Y <= y) b) fx(x) c) fy(y) d) Are X and Y independent? Give a reason for your answer.The random vector (X, Y ) has the following joint probability density function:f(X,Y )(x, y) = 4xye^−(x2+y2), x > 0, y > 0,0 , otherwiseLet Z =√(X^2 + Y ^2) . Find the probability density of the random variable Z.For random variables X and Y with joint density function f(x,y) = 6e^-2x-3y. (x,y > 0) and f(x,y) = 0 otherwise, find: Are X and Y independent? Give a reason for your answer.
- Consider the random variable X with PDFf(x) = e−x / (1 + e−x)2 , x ∈ R.Find the density function of Y = 1/ (1 + e−X)The joint density function of X and Y is given by f(x, y) = xe-(x+y) x > 0, y > 0 0 otherwise What is P(x < 1, y < 1)?Let f(x, y) = C/8, 0 ≤ y ≤ 4, y ≤ x ≤ y + 2, be the joint pdf of X and Ya) Find the value of C so that f(x,y) is a valid joint p.d.fb) Find the marginal probability density function of X, fX(x)c) Compute μX , μY , Cov(X, Y), and ρ.