3) The joint density of X and Y is f( x, y) = 2e *2) for x > 0, y > 0 and f( x, y) 0 otherwise. a) Find the joint cumulative distribution function. b) Find P( X
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- Suppose a continuous random variable X~Fx(x): f(x,y) = {1/4e^-1x/4, if x≥0 0, x<0} What is the cumulative density function of Y=min{2,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 Find the constant c, P(Y≥1/2), P(X < 2, Y >1/2), P(X < 1), Determine whether X and Y are independent.If X and Y are independent exponential random variables, each having parameter λ.(a) Find the joint density function of U = X + Y by using the convolution of fX and fY .(b) Find the joint density function of V = X − Y by using the method of transformation.(c) Are U and V independent?
- If two random variables X1 and X2 have the joint density function given by f (x1, x2) = x1x2, 0 < x1 < 1, 0 < x2 < 2 0, otherwise Find the probability that (a) Both random variables will take on values less than 1 (b) The sum of the values taken on by the two random variables will be less than 1.Find the density of Z = X-Y for independent Exp(λ) random varibles X and Y. Show all Work!!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 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 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 given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),Let (X, Y) be a uniformly distributed random point on the quadrilateral D with vertices (0,0), (2,0), (1,1) and (0,1). (a) Find the joint density function of (X; Y ) and the marginal density functions of X and Y . (b) Find E[X] and E[Y ]. (c) Are X and Y independent?
- The time (in minutes) required to complete a certain subassembly is a random variable X with the density function f(x) = 1/21 x^2, 1 ≤ x ≤ 4.(a) Use f (x) to compute Pr(2 ≤ X ≤ 3).(b) Find the corresponding cumulative distribution function F(x).(c) Use F(x) to compute Pr(2 ≤ X ≤ 3).Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)The random vector (X,Y) has the following joint probability density function:f(X,Y)(x,y) ={4xye−(x^2+y^2), x >0, y >0, 0, otherwise LetZ=√(X2+Y2) Find the probability density of the random variableZ.