# Question

Define a random process according to

X[n] = X [n– 1] + Wn , n = 1, 2, 3, …

Where X [0] = 0 and Wn is a sequence of IID Bernoulli random variables with and Pr( Wn = 1)= p and Pr( Wn = 0) = 1 – p.

(a) Find the PMF, PX (k; n) = Pr (X[k] = n).

(b) Find the joint PMF, PX1, X2 (k1, k2 ; n1, n2) = Pr (X [k1] = n1, X [k2] = n2).

(c) Find the mean function, µX [n] = E [X[n]].

(d) Find the autocorrelation function, RX, X [k, n] = E [X [k] X [n]].

X[n] = X [n– 1] + Wn , n = 1, 2, 3, …

Where X [0] = 0 and Wn is a sequence of IID Bernoulli random variables with and Pr( Wn = 1)= p and Pr( Wn = 0) = 1 – p.

(a) Find the PMF, PX (k; n) = Pr (X[k] = n).

(b) Find the joint PMF, PX1, X2 (k1, k2 ; n1, n2) = Pr (X [k1] = n1, X [k2] = n2).

(c) Find the mean function, µX [n] = E [X[n]].

(d) Find the autocorrelation function, RX, X [k, n] = E [X [k] X [n]].

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