# Question: Suppose that Xk is a sequence of IID Gaussian random

Suppose that Xk is a sequence of IID Gaussian random variables. Recall that the sample variance is given by

(a) Show that the sample variance can be written as a quadratic form ŝ2 = XTBX and find the corresponding form of the matrix B.

(b) Use the techniques outlined in Section 6.4.2 to show that the characteristic function of ŝ2is

(c) Show that the PDF of ŝ2is that of a chi- square random variable.

(a) Show that the sample variance can be written as a quadratic form ŝ2 = XTBX and find the corresponding form of the matrix B.

(b) Use the techniques outlined in Section 6.4.2 to show that the characteristic function of ŝ2is

(c) Show that the PDF of ŝ2is that of a chi- square random variable.

## Answer to relevant Questions

Suppose is a vector of IID random variables where each element has some PDF, fX (x). Find an example PDF such that the median is a better estimate of the mean than the sample mean. Show that if Xn ,n = 1, 2, 3, … is a sequence of IID Gaussian random variables, the sample mean and sample variance are statistically independent. Let X (t) = A(t) cos (ω0t + θ), where A(t) is a wide sense stationary random process independent of θ and let θ be a random variable distributed uniformly over . Define a related process Y (t) = A (t) cos((ω0 +ω1) t + ...Consider a discrete- time wide sense stationary random processes whose autocorrelation function is of the form Assume this process has zero- mean. Is the process ergodic in the mean? Let X (t) be a wide sense stationary Gaussian random process and form a new process according to Y (t) = X (t) cos (ωt + θ) where ω and θ are constants. (a) Is Y (t) wide sense stationary? (b) Is Y (t) a Gaussian ...Post your question