(d) Find the conditional density function of Y, given Y, = Y1. , where p(Y2=Y -). (Enter your probability: (e) Find a fraction.)
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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
- 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, ..., 2Can you provide a detailed explanation on solving the following problem? Let X and Y have joint probability density funtion :f(x, y) = x + y , for 0 < x < 1 and 0 < y < 1. 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]
- 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).The random variables X and Y have the following joint probability density function:f(x,y)={e−x−y , 0<x<∞; 0, elsewhere. What is Cov(X,Y)(X,Y)?Find a value of k that will make f a probability density functionon the indicated interval.ƒ(x) = kx; [0, 3]
- 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 the value of the constant b so that the function f_x, y(x, y) = bxy^2 exp(-2xy)u(x-2)u(y - 1) is a valid joint probability densitySuppose 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?