7) (20 pts) Let X(t) = At be a random process, such that A is N(0,...
Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events (think of incoming phone calls, customers entering a bank, car accidents, etc.) that occur up to and including time t where the occurrence times of these events satisfy the following three conditions Events only occur after time 0, i.e., X(t)0 for t0 If N (1, 2], where 0< t t2, denotes the number of events that occur in the time interval (t1,...
6) (6 pts) Let X, X, and X; be a random sample (n = 3) from a population with mean u and standard deviation o. Consider two estimators of u: T1 = (X1 + X2 + X3)/3 and T, = 0.10 X2 +0.25 X. + 0.65 X. Recall that because 71 is the sample average, E(71) - u and Var(T) = Oʻ/3. (a) (3 pts) Find the expected value and variance of T2. (b) (3 pts) Would T, or T2...
Consider a random process X(t) defined by X(t) - Ycoset, 0st where o is a constant 1. and Y is a uniform random variable over (0,1) (a) Classify X(t) (b) Sketch a few (at least three) typical sample function of X(t) (c) Determine the pdfs of X(t) at t 0, /4o, /2, o. (d) EX() (e) Find the autocorrelation function Rx(t,s) of X(t) (f) Find the autocovariance function Rx(t,s) of X(t) Consider a random process X(t) defined by X(t) -...
Problem 3 Consider the Gaussian process, X(t), with zero mean and a utocorrela- t ) i,2 tion function Rx(t1, t2 mini 1. Find the covariance matrix of the random variables X(1) and X (2) 2. Write an expression for the joint PDF of X(1) and X(2) Problem 3 Consider the Gaussian process, X(t), with zero mean and a utocorrela- t ) i,2 tion function Rx(t1, t2 mini 1. Find the covariance matrix of the random variables X(1) and X (2)...
P9.3 A random process X(t) has the following member functions: x1 (t) -2 cos(t), x2(t)2 sin(t), x3(t)- 2 (cos(t) +sin(t)),x4t)cost) - sin(t), xst)sin(t) - cos(t).Each member function occurs with equal probability. (a) Find the mean function, Hx (t). (b) Find the autocorrelation function, Rx(t1,t2) (c) Is this process WSS? Is it stationary in the strict sense?
2. Consider the random process x(t) defined by x(t) a cos(wt 6), where w and 0 are constants, and a is a random variable uniformly distributed in the range (-A, A). a. Sketch the ensemble (sample functions) representing x(t). (2.5 points). b. Find the mean and variance of the random variable a. (5 points). c. Find the mean of x(t), m(t) E((t)). (5 points). d. Find the autocorrelation of x(t), Ra (t1, t2) E(x (t)x2 )). (5 points). Is the...
Let X(t) and Y(t) be independent, wide-sense stationary random process with zero means and the same covariance function Cx(t) Let Z(t) be defined by Z(t) = X(t)coswt + Y(t)sinwt Find the joint pdf of X(t1) and X(t2) in part b
1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectively. Obtain the conditionafdistributiono) Moreover, find EX Y X2t t given Yt-n, n = 1,2. 2, (a) Let T be an exponential random variable with parameter θ. For 12 0, compute (b) When Amelia walks from home to work, she has to cross the street at a certain point. Amelia needs a gap of a (units of time) in the traffic to cross the...
3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with common power spec- tral density given by SX (f) = SX (f) = 112B(f) watt/Hz. Define x(t) = Xe(t) cos(2tht)--Xs(t) sin(2tht) where fo 》 (a) Is X(t) a Gaussian process? (b) Find the mean E(X (t), autocorrelation function Rx (t,t + T), and power spectral density Sx(f) of the process X(t) (c) Find the pdf of X(O) (d) The process X(t) is passed through...
Exercise 5. Let X(t) be a WSS process with correlation function 1-Irl, if-1-1S1 0,otherwise. Rx(T) = It is known that when X (t) is input to a system with transfer function H(), the system output Y(t) has a correlation function Ry(T) sin TT = =-TT Find the transfer function H(u