0〈z,0〈y Given the following joint distributionfrY(x,y)-, cez+2y else Calculate the following 1. The value of c...
Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint pdf given by fx,YZ(xgz) = k xyz if 0 S$ 1,0 rS 1,0 25 1 ) and fxyZ(x,y,z) = 0, otherwise. (a) Find k so that fxyz(x.yz) is a genuine probability density function. (b) Are X,Y,Z independent? (c) Find PXs 1/2, Y s 1/3, Z s1/4). (d) Find the marginal pdf fxy(x.y). (e) Find the marginal pdf fx(x). Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint...
1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a. Find the joint pdf fxy(x, y), marginal pdf (fx(x) and fy()) and cdf (Fx(x) and Fy)) b. Find the conditional pdf fxiy Cr ly c. Find the probability P(X < Y = y) d. Are X and Y independent?
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
4. Two random variables X and Y have the following joint probability density function (PDF) Skx 0<x<y<1, fxy(x, y) = 10 otherwise. (a) [2 points) Determine the constant k. (b) (4 points) Find the marginal PDFs fx(2) and fy(y). Are X and Y independent? (c) [4 points) Find the expected values E[X] and EY). (d) [6 points) Find the variances Var[X] and Var[Y]. (e) [4 points) What is the covariance between X and Y?
(II) Multiple continuous random variables: 8.2 Let X and Y have joint density fXY(x,y) = cx^2y for x and y in the triangle defined by 0 < x < 1, 0 < y < 1, 0 < x + y < 1 and fXY(x,y) = 0 elsewhere. a. What is c? b. What are the marginals fX(x) and fY(y)? c. What are E[X], E[Y], Var[X] and Var[Y]? d. What is E[XY]? Are X and Y independent?
Let the random variables X, Y with joint probability density function (pdf) fxy(z, y) = cry, where 0 < y < z < 2. (a) Find the value of c that makes fx.y (a, y) a valid pdf. (b) Calculate the marginal density functions for X and Y (c) Find the conditional density function of Y X (d) Calculate E(X) and EYIX) (e Show whether X. Y are independent or not.
7. The random variables X and Y have joint probability density function f given by 1 for x > 0, |y| 0 otherwise. 1-x, Below you find a diagram highlighting the (r, y) pairs for which the pdf is 1 (a) Calculate the marginal probability density function fx of X (b) Calculate the marginal cumulative distribution function Fy of Y (c) Are X and Y independent? Explain.
7. The random variables X and Y have joint probability density function f given by 1 for x > 0, |y| 0 otherwise. 1-x, Below you find a diagram highlighting the (r, y) pairs for which the pdf is 1 (a) Calculate the marginal probability density function fx of X (b) Calculate the marginal cumulative distribution function Fy of Y (c) Are X and Y independent? Explain.
Suppose that X and Y are random variables the following joint PDF: fxy(x,y) = otherwise Determine fx, the marginal PDF of X. a. etermine Fx, the marginal CDF of X.
a. Given the joint probability den- sity function fxy(x, y) as, Skxy, (x, y) e shaded area Jxy(, 9) = 10 otherwise Find [i] k [ii] fx(x) [iii] fy(y) Are X and Y independent? b. Given the joint probability density function fxy(x, y) as, fxy(x, y) = { 0 kxy, (x, y) E shaded area otherwise Find [i] k [ii] fx(x) [iii] fy(y) Are X and Y independent? 2 1