If a random process x(t) has no periodic components and if x(t) is of non-zero mean then Ryy (T) = [E(X)]² %3D
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![If a random process x(t) has no periodic components
and if x(t) is of non-zero mean then Ry (T) = [E(X)]²
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.A stationary unity mean random process X (t) has the auto correlation function RYy (T) = 1 + e 2, Find the mean and variance ofY= 1 x (t) · dt2 Let X (t) be a random process with mean 3 and auto correlation R(t, t2) = 9 + 4 e-0.2 ,-t, Determine the mean, variance and covariance of the random variables Z =X (5) and wX (8).
- A Random process X(t) is applied to a network with impulse response h(t) = u(t) exp (-bt) where b > 0 is a constant. The cross- correlation of X(t) with the output Y(t) is known to have the same form, Ryy(t) = u(t)t exp(-bt) (i) Find the auto-correlation of Y(t).A random process X(t) is stationary. If it is known that E[X(10)] = 10 and var(X(10)) = 1, then determine EX(1)] and var (X(1)). E[X(1)]: [ var(X(1)): [Suppose X is a random variable and h is a non-increasing function, i.e., h(x1) #2. Please show that h(X) and h(-X) are negatively correlated, that is Cov[h(X), h(-X)] <0.
- Let N(t) be the percentage of a state population infected with a flu virus on week t of an epidemic. What percentage is likely to be infected in week 4 if N(3) = 8 and N'(3) = 1.2?Suppose Xn is an IID Gaussian process, withµX[n]=1, and σ2 X[n]=1Now, another stochastic process Yn = Xn − Xn−1. Please find:(a) The mean µY (n).(b) The variance σ2Y (n).(c) The auto-correlation RY (n, k)4 random process X (t) has an auto correlation function RYx (t) = A2 + B. e-lt, where A and B are positive constants. Find the mean value of the response of a system having an impulse respònse h (t) = e-kt for t > 0 = 0 for t< 0 . where K is a real positive constant, for which X (t) is its input.
- Éxample 10: Show that the random process X (t) = A e'at is WSS if and only if E [A] = 0.5- Suppose that { W(t): t 0} is a Brownian motion process with variance ow = 3. Find P{ W(1) < 1| W(2) = 5/4 } = ?Consider the random process W(t) = X cos(2π fot) + Y sin(2π fot) where X and Y are uncorrelated random variables, each with expected value 0 and variance o². (a) Find the auto-correlation function of the random process W(t). (b) Is W (t) wide sense stationary (WSS) ?