9-A-Two random variables X and Y are independent and have marginal Probability Density Functions (PDF) shown below....
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.
2. Suppose X and Y are independent random variables with the pdf (probability density func- tion) f(x) e-2 for x > 0. (a) What is the joint probability density function of (X, Y)? (b) Define W-X-Y, Z = Y, then what is the Joint probability density function fw.z(w, z) for (W, Z). (c) Determine the region for (w, z) where fw.z is positive. (d) Calculate the marginal probability density function for W.
2. Suppose X and Y are independent random variables with the pdf (probability density func- tion) f(x)- for x > 0. (a) What is the joint probability density function of (X, Y)? (b) Define W = X-Y, Z = Y, then what is the joint probability density function fw,z(w, z) for (W, Z). (c) Determine the region for (w, z) where fw,z is positive. (d) Calculate the marginal probability density function for W
2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector, 2) Two statistically-independent random variables, (X,Y), each have marginal probability density, N(0,1) (e.g., zero-mean, unit-variance Gaussian). Let V-3X-Y, Z = X-Y Find the covariance matrix of the vector,
Let X and Y be two random variables with the joint probability density function: f(x,y) = cxy, for 0 < x < 3 and 0 < y < x a) Determine the value of the constant c such that the expression above is valid. b) Find the marginal density functions for X and Y. c) Are X and Y independent random variables? d) Find E[X].
(45) Two random variables X and Y have the joint probability density | 2, 0sxs1 and 0 s ys1 and x + y21 fxY (x, y) = 0, elsewhere Answer each of these independent questions about X, Y, carefully indicating the domain of all functions where needed. Parts a). - i). are 5 points each. a). Find E(Z), where Z is a new random variable defined by Z = XY b). A is the event {X >0.75}. Find P(A). c)....
Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x, 0<x<1 (zero otherwise), and g(y) = 3y2, 0<y<1 (zero otherwise). Let U = min(X,Y), and V = max(X,Y). Find the joint pdf of U and V.
The joint probability density function of the random variables X, Y, and Z is (e-(x+y+z) f(x, y, z) 0 < x, 0 < y, 0 <z elsewhere (a) (3 pts) Verify that the joint density function is a valid density function. (b) (3 pts) Find the joint marginal density function of X and Y alone (by integrating over 2). (C) (4 pts) Find the marginal density functions for X and Y. (d) (3 pts) What are P(1 < X <...
Consider the following joint probability density function of the random variables X and Y : 3x−y , 1 < x < 3, 1 < y < 2, f(x, y) = 9 0, elsewhere. (a) Find the marginal density functions of X and Y . (b) Are X and Y independent? (c) Find P(X > 2).
The joint probability density function (PDF) of random variables X and Y is given by: f(x,y) = 4xy for 0 ≤ y ≤ x ≤ 1, and = 0 elsewhere The mean of the random variable X is: