# Question: Define a random process according to X n X n 1

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]].

## Relevant Questions

Consider a Poisson counting process with arrival rate, λ. Suppose it is observed that there have been exactly arrivals in [0, t] and let S1, S2… Sn be the times of those arrivals. Next, define X1, X2… Xn to be a ...In this problem, we develop an alternative derivation for the mean function of the shot noise process described in Section 8.7, Where the Si are the arrival times of a Poisson process with arrival rate, λ, and h (t) is an ...A random process X (t) has the following member functions: x1 (t) = – 2cos (t), x2 (t) = – 2sin(t) x3 (t) = 2[cos( t) + sin(t)] x4(t) = [cos (t) – sin(t)]. Each member function occurs with equal probability. (a) Find ...Two students play the following game. Two dice are tossed. If the sum of the numbers showing is less than 7, student A collects a dollar from student B. If the total is greater than 7, then student B collects a dollar from ...A random waveform is generated as follows. The waveform starts at 0 voltage. Every seconds, the waveform switches to a new voltage level. If the waveform is at a voltage level of 0 volts, it may move to + 1 volt with ...Post your question