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- 5. Suppose that the joint probability density function of X and Y is given by f(x, y) = C (x² + y), 0 < x < 2, 0 < y < 2 and f(x, y) = 0 otherwise. (a) Find C. (b) Find the marginal density functions of X and Y. (c) Compute P(X > Y). (d) Compute E[X]. (e) Compute E[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?1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.
- Suppose that the joint probability density function of X and Y is fX,Y(x,y) = 10.125(x2 – y2) e−3x , for 0<x<∞ and -x<y<x 0, otherwise Give your answers to the below questions in two decimal places where appropriate. (a) The marginal probability density function of X is given by: fX(x) = A xB e-3x , for 0<x<∞ 0, otherwise Find the value of A. (b) Find the value of B. (c) The conditional probability density function of Y, given that X=x for some x>0, takes the following form: fY|X=x(y) = C (x2-y2) xD e-Ex, for -x<y<x 0, otherwise. Find the value of C (d)Find the value of D. (e)Find the value of E.Let the joint Probability Density Function of X and Y be: 2*e(x+y) for -infinity<x<y<1. Find the Cov(X, Y) and find the Marginal Probability Density Functions of X and Y1. Let f (x) = cx 3 and S X = [0, 2] (X is continuous and c is a constant). a. What value of c will make f (x) a valid density?b. What is P (X = 1)?c. What is P (X = 1 or X = 2)?
- Let X and Y have the joint PDF f(x,y) = 2e−x−y for 0<x<y<∞. Find the joint PDF of Z = 2X and W = Y − X, as well as the two marginal PDFs.Suppose that X, Y are jointly continuous with joint probability density function f( x, y){ xe^-x(1+y), ifx >0 and y >00, otherwise. (a) Find the marginal density functions of X and Y. (b) Calculate the expectation E[XY]. (c) Calculate the expectation EIX/(1+ Y )1. (e) Determine if the random variables X and Y in this exercise are independent.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
- a. What is the joint marginal PDF of X and Z?b. What is [P 0 <= Y <= 3/2 | X = 1/3 Z = 5/2]?Suppose that X and Y have the following joint probability density function.f (x, y) = 3x 400 0 < x < 6, y > 0, x − 4 < y < x + 4 (a) Find E(XY). (b) Find the covariance between X and Y.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?