Let X(t) X(t) be a Gaussian random process with μ X (t)=0 μX(t)=0 and R X ( t 1 , t 2 )=min( t 1 , t 2 ) RX(t1,t2)=min(t1,t2) .
Find P(X(4)<3|X(1)=1) P(X(4)<3|X(1)=1) .
Let X(t) X(t) be a Gaussian random process with μ X (t)=0 μX(t)=0 and R X...
7) (20 pts) Let X(t) = At be a random process, such that A is N(0, 1). , ??(t)-EX(t)]. (a) Find mean of the random process X(t) (b) Find the auto-correlation function of X, Rx(t1,t2) - E[X (ti)X (t2)
7) (20 pts) Let X(t)-At be a random process, such that A is N(0, 1). (b) Find the auto-correlation function of X, Rx(t1, t2) E[X(ti)X(t2)
Autocorrelation of an X(t) random process is Rxx (t1, t2) = 4e-t-t2 This a Gaussian process with mean zero. a) [6p] Is this process wide sense stationary? Briefly explain. b) [9p] Calculate the probability P (X(2)> 1) using the Table at the cover. c) [10p] Calculate approximately the probability P(X(2) > X(4) + 1). Some useful relations 1. Var(X(t)) = E({€)) - (E(X(t))) 2. R(X(t)X(t) = ELX(t-)X(02)]| 3. Var(X(c) +X)) = Var( (t) ) + Var (X (t2) - 2Cov(X...
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)...
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inean ,, and variance σ2, namely, fx (x) = exp ( 2-2 . Let X and Y be two i.i.d. random variables distributed as Gaussian with mean 0 and variance 1. Show that Z-XY is again a Gaussian random variable but with mean 0 and variance 2. Show your full proof with integrals. 2. From above, can you derive what will be 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...
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...
324. Consider the random process X(t) = A + Bt2 for - <t < oo, where A and B are two statistically independent Gaussian random variables, each with zero mean and variance o?. a) Plot two sample functions of X(t). b) Find E{X(0)} c) Find the autocorrelation function Rx(t,t +T). d) Find the pdf of the random variable Y = X(1). e) Is X(t) a Gaussian process? Prove your result.
The random process X(t) is defined by X(t) = X cos 27 fot + Y sin 2 fot, where X and Y are two zero-mean Gaussian random variables, each with the variance 02. (a) Find ux(t) (b) Find RX(T). Is X(t) stationary? (c) Repeat (a) and (b) for 0 + 0
1) Random Processes: Suppose that a wide-sense stationary Gaussian random process X (t) is input to the filter shown below. The autocorrelation function of X(t) is 2xx (r) = exp(-ary Y(t) X(t) Delay a) (4 points) Find the power spectral density of the output random process y(t), ΦΥΥ(f) b) (1 points) What frequency components are not present in ΦYYU)? c) (4 points) Find the output autocorrelation function Фуу(r) d) (1 points) What is the total power in the output process...