ECE331 HOMEWORK 7

For this homework X_n means X with subscript n
and a^n means a to the power n.

(0)[reading corresponding with the lectures] 
Read Yates chapter 6.1-6.4, 6.7-6.9 

(1) Define a discrete time stochastic process X_n as follows.
A fair coin is tossed once.
If the outcome is heads then X_n=1 for all n.
If the outcome is tails then X_n=-1 for all n.
(Please note that this is a very simple and trivial stochastic process.)
(a) sketch some sample paths for the process
(b) Find the pmf of X_n
(c) Find the joint pmf of X_n and X_(n+k)
(d) Find the mean and autocovariance functions of X_n

(2) Redo (1) with the following change:
A fair coin is tossed.  
If the outcome is heads then X_n=(-1)^n for all n.
If the outcome is tails then X_n=(-1)^(n+1) for all n.

(3) Consider the moving average processes
Y_n=(X_n+X_(n-1))/2 and
Z_n=(2 X_n+X_(n-1))/3.
Let X_n be a Bernoulli stochastic process with probability p=1/2.
(a) Find the mean, variance, autocovariance and autocorrelation of X_n.
(b) Find the mean, variance and autocovariance of Y_n.
(c) Find the mean, variance and autocovariance of Z_n.
(d) Flip a coin 10 times to obtain a realization of
X_n; n=1,2,...,10. Find the resulting realizations of Y_n and Z_n.
Are the sample means of X_n, Y_n, Z_n close to their
means computed above?

(4) Consider the autoregressive processes
Y_n=2 Y_(n-1)+X_n;   Y_0=0, and
Z_n=(1/2) Z_(n-1)+X_n;   Z_0=0.
Let X_n be a Bernoulli stochastic process with probability p=1/2.
(a) By taking the expectation of the equations for Y_n and Z_n,
and solving the resulting linear systems,
find EY_n and EZ_n.
(b) Flip a coin 10 times to obtain a realization of
X_n; n=1,2,...,10. Find the resulting realizations of Y_n and Z_n.
Relate the behaviors in (a) and (b).

(5) Let A_n be a Bernoulli random process with p=1/4 and
let X_n=f(A_n), where f(0)=-1 and f(1)=1.
Note that X_n are iid.
Consider the sum process
S_n=S_(n-1)+X_n;   S_0=0
Please find the following:
(a) ES_n
(b) Variance of S_n
(c) Autocovariance of S_n
(d) Autocorrelation of S_n
(e) distribution of S_n
(f) joint distribution of S_4 and S_7
(g) joint distribution of S_n and S_m, where m>n

(6) 
(a) Redo question 5 with X_n iid and X_1 distributed as N(0,1)
(b) Redo question 5(a)(b)(c)(d)(e) with X_n iid and X_1 distributed as N(2,1)