Problem 2 (15pts). Consider the following joint density function 0, else (a) Find the conditional density...
Consider the joint density function f(x, y) = else (a) Find the marginal density functions for X and Y (b) Compute P(Y 亻1/2/X 3/4). (c) Find the conditional density function X given Y = y. (d) Compute P(Y 1/2lX-3/4).
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
Problem 3 (15pts). Let X and Y have joint pdf 0. else Find the pdf of Z A +Y
Consider fx (x)=e*, 0<x and joint probability density function fx (x, y) = e) for 0<x<y. Determine the following: (a) Conditional probability distribution of Y given X =1. (b) ECY X = 1) = (c) P(Y <2 X = 1) = (d) Conditional probability distribution of X given Y = 4.
(+5) Answer the following for the joint density function analyzed in class: [0,2],y E [0, 1] f(x,y) = if else (a) (+2) The conditional density fy(y|X = 1). (b) (+1) The conditional mean E(Y|X = 1). (c) (+2) The conditional standard deviation ơyIXe1.
Consider a continuous random vector (Y, X) with joint probability density function F(x,y) = e-y for 0<x<y<∞ Compute the marginal density of X denoted by f(x). Compute the conditional density of Y given X denoted by f(y|x). Hint: Consider the two cases y > x and y ≤ x separately. Compute the conditional expectation E[Y |X = x]. Compute the conditional variance Var(Y |X = x).
4. The random variables X and Y have joint probability density function fx.y(r, y) given by: else (a) Find c (b) Find fx (r) and fr (u), the marginal probability density functions of X and Y, respectively (c) Find fxjy (rly), the conditional probability density function of X given Y. For your limits (which you should not forget!), put y between constant bounds and then give the limits for r in terms of y. (d) Are X and Y independent?...
012) e yi 0, elsewhere. (a) Verify that the joint density function is valid. (2 points) (b) Find P(Y, < 2,Y2 > 1). (2 points) (c) Find the marginal density function for Y2. (2 points) (d) What is the conditional density function of Yi given that Y2-?2 points) (e) Find P(Y > 2|Y 1). (2 points)
Let the joint density function of X and Y be given by the following x +y for 0 < x < 1 and 0 < y < 1 f(x, y) = 0 otherwise Find E[X], E[Y], Var[X], Var[Y], Cov(X,Y), and px,y Find E[X]Y], E[E[X|Y]], and Var[X|Y]. Find the moment generating function Mx,y(t1, t2)
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distribution of X given Y-0. (i.e., Pxjy (xY-0)) (c) Compute EXIY-01 and Var(X)Y = 0). (d) Find the covariance between X and Y. (i.e., Cov(X, Y)) (e) Are X and Y independent? Justify your answer.
(20 points) Consider the following joint distribution of X and...