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7.6.2 X and Y are jointly Gaussian ran- dom variables with E[X] and Var[X] = Var[Y]...
2. Let X and Y be jointly Gaussian random variables. Let ElX] = 0, E[Y] = 0, ElX2-4. Ey2- 4, and PXY = [5] (a) Define W2x +3. Find the probability density function fw ( of W. [101 (b) Define Z 2X - 3Y. Find P(Z > 3) 5] (c) Find E[WZ], where W and Z are defined in parts (a) and (b), respectively.
2. (30 Points) X and Y ~ N (0,4) are two jointly Gaussian random variables, and E(XY) = 3 a. (10 Points) Find their joint PDF, f (x,y). b. (10 Points) Find the mean and variance of Z = X +Y. c. (10 Points) Find the mean and variance of Z = X + Y + 2.
4. Assume that the random variables X and Y are jointly Gaussian but are not statistically independent. Suppose that X has (90,4), Y has (75,5), and ρ--025 Express the joint pdf of the two random variables.
1. Two normal random variables X and Y are jointly distributed with Var(X) 25 and Var(Y) 1600. It is known that P(Y>80| X = 50) 0.1 and P(Y 22 X 40) 0.7886 (1) What is the correlation coefficient between X and Y? (2) What is the expected value of Y given X 50?
For the random variables X and Y having E(X) = 1, E(Y) = 2, Var (X) = 6, Var (Y) = 9, and Pxy = -2/3. Find a) The covariance of X and Y. b) The correlation of X and Y. c) E(X2) and E(Y2).
Suppose that X, Y, and Zare jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(x)-4 E(Y) 2 E(Z)-7 Var (x) -28 Var(Y)-3 Var (Z) -44 Compute the values of the expressions below. E(Y -1) 5Z + 4X Var (4Y-3)
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(x)-4 E(Y) 2 E(Z)-7 Var (x) -28 Var(Y)-3 Var (Z) -44 Compute the values of the expressions below. E(Y 1) 5Z + 4x Var (4Y-3)
(12 points) The random variables X1, X2, and X; are jointly Gaussian with the following mean vector and covariance matrix: 54 2 07 2 5 -1 0-1 The random variable Y is formed from X1, X2, and X; as follows: Y=X1 - X2 + X3 +4. Determine P( Y> 3).
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(X)-8 E(Y)-7 E(Z)-2 Var (x) 24 Var (Y) 2 Var (z) 29 Compute the values of the expressions below. E (5x- 4) Var (-2 5z) - [D
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(x) = 3 Var (x) = 19 E(r)--2 Var (r)-36 E(Z)-6 Var (Z)-45 Compute the values of the expressions below. E (2 1)