For the digital filter of Problem 11.2.6, generate 100 sample paths Y₀,..., Y₅₀₀ assuming the X_i are iid Gaussian (0, 1) random variables and Y(0) = 0. Estimate the expected value and variance of Y_n for n = 5, n = 50 and n = 500.

Problem 11.2.6

The input ..., X_-1, X₀, X₁,... and output ..., Y_-1, Y₀, Y₁,... of a digital filter obey

Let the inputs be a sequence of iid random variables with E[X_i] = µ_Xi = 0 and Var[X_i] = σ². Find the following properties of the output sequence: E[Y_i], Var[Y_i], Cov[Y_i+1, Y_i], and ρ_Yi+1,Y_i.