I just want to check my answer before I submitted it.

The random process X (0:33" is a family of exponentials dependending on the random variable a. (a) Draw three sample functions of this random process. (b) Express the mean 77(1'), the autocorrelation R(t1,t ), and the rst-order density f (x,t) of X (t) in terms of the density fa(a) of a. 6. The random variable c is uniform in the interval (0,1). Find R(t1,t2) if (a) X (t)=u(tc)- (b) Y(t)=6(tc)- 7. 2(1):111(t)cos(a)ct+6)+n2(t)sin(a)ct+6), where (9 is uniformly distributed in (0, 272'); n1(t) and n20) are WSS (wide-sense-stationary); 1210'), I120) and (9 are all independent. (a) Find E(Z(t)}. (b) Find R,(t,t2) . Is Z(t) WSS? (c) Find R, (0), if Z(t) is WSS. (Video is required.) 8. Poisson Process is defined by the following properties: (1) The number n(t, t,) of the points t; in the interval (t, t,) of length t=t,-t, is a Poisson random variable with parameter it : P(n(t , t, ) = k) = e-at (at) k k ! (2) If the intervals (t, t, ) and (t,, t) are non-overlapping, then the random variables n(t,, t,) and n(t,, t,) are independent. (3) At a time moment t, there is no more one Poisson point. Using the points t;, we form the stochastic process X(t)=n(0, t).Poisson process '. I (One sample) . ', Find (aTE{X(r)}- (b) Rap r2). (Video is required.) 9. Suppose that X (t) is a Poisson process as in the above question such that E{X(9)} =6. (a) Find the mean and the variance of X (8) . (b) Find P(X(2) s 3)