Random process- Probability
A stochastic process X(t) is defined via: X(t,w) = A(w)t + Bw), te 1-1, 1], where Aw) ~ U([-1,1]) and B(w) ~ U((-1,1]) are statistically independent random variables. For this process: 2.a) plot two sample realizations x1(t) and x2(t). 2.b) Determine the first-order PDF fx(x;t) associated with it. 2.c) Determine the mean pz(t) and variance ož(t). 2.d) Determine the autocorrelation Rex(ti, t2) and the auto-covariance Cxx(t1, t2) associated with it.
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7) (20 pts) Let X(t) = At be a random process, such that A is N(0,...
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)...
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)
Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events...
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,...
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...
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...
Problem 3 Consider the Gaussian process, X(t), with zero mean and a utocorrela- t ) i,2 tion function Rx(t1, t2 mini 1....
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),...
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?
Let X(t) and Y(t) be independent, wide-sense stationary random process with zero means and the same covariance function...
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
Autocorrelation of an X(t) random process is Rxx (t1, t2) = 4e-t-t2 This a Gaussian process...
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...
Let X(t) be a wide-sense stationary random process with the autocorrelation function :
Let X(t) be a wide-sense stationary random process with the autocorrelation function : Rxx(τ)=e-a|τ| where a> 0 is a constant. Assume that X(t) amplitude modulates a carrier cos(2πf0t+θ), Y(t) = X(t) cos(2πf0t+θ) where θ is random variable on (-π,π) and is statistically independent of X(t). a. Determine the autocorrelation function Ryy(τ) of Y(t), and also give a sketch of it. b. Is y(t) wide-sense stationary as well?
The sample function X(t) of a stationary random process Y(t) is given by X(t) = Y(t)sin(wt+Θ) whe...
The sample function X(t) of a stationary random process Y(t) is given by X(t) = Y(t)sin(wt+Θ) where w is a constant, Y(t) and Θ are statistically independent, and Θ is uniformly distributed between 0 and 2π. Find the autocorrelation function of X(t) in terms of RYY(τ).
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)
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,...
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...
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?
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...
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