# Question: Suppose X Y and Z are independent zero mean unit

Suppose, X, Y, and Z are independent, zero- mean, unit- variance Gaussian random variables.

(a) Using the techniques outlined in Section 6.4.2, find the characteristic function of

W = XY + XZ + YZ.

(b) From the characteristic function found in part (a), find the mean and variance of .

(c) Confirm your answer in part (b) by finding the mean and variance of W directly. In this part, you may want to use the result of the Gaussian moment factoring theorem developed in Exercise 6.18.

(a) Using the techniques outlined in Section 6.4.2, find the characteristic function of

W = XY + XZ + YZ.

(b) From the characteristic function found in part (a), find the mean and variance of .

(c) Confirm your answer in part (b) by finding the mean and variance of W directly. In this part, you may want to use the result of the Gaussian moment factoring theorem developed in Exercise 6.18.

## Answer to relevant Questions

Find the PDF of Z = X1X2 + X3X4 + X5X6 + X7X8 assuming that all of the Xi are independent zero- mean, unit- variance, Gaussian random variables. Hint: Use the result of Special Case # 2 in Section 6.4.2.1 to help. Suppose a point in two- dimensional Cartesian space, (X, Y), is equally likely to fall anywhere on the semicircle defined by X2 + Y2 = 1and Y ≥ 0. Find the PDF of Y, fY (y). The traffic managers of toll roads and toll bridges need specific information to properly staff the toll booths so that the queues are minimized (i. e., the waiting time is minimized). (a) Assume that there is one toll ...Let be a pair of independent random variables with the same exponential PDF, fXi (x) = exp(– x) u( x) i = 1, 2 Define Y1, Y2 to be the order statistics associated with the Xi. That is, Y1 = min (X1, X2) and Y2 = min (X1, ...Consider the random sequence Xn = X / (1 + n2), where is a Cauchy random variable with PDF, Determine which forms of convergence apply to this random sequence.Post your question