(4) If two random processes X(1) and Y(t) are statistically independent and are atleast wide sense stationary, then Ryy (7) = XY
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.1. Suppose that Y₁ = A + Bt + Xt, where {Xt} is a random walk. First suppose that A and B are constants. (a) Is {Y} stationary? (b) Is {VY} stationary? Now suppose that A and B are random variables that are independent of the random walk {Xt}. (c) Is {Y} stationary? (d) Is {VY} stationary?(b) Let N(1) be a a zero-mean white Gaussian noise random process. Define V = N(t)dt. Find mean and variance of V.
- Consider a random process x(t) = Acos(@t+0), where and are constants and 4 is a random variable with zero mean and variance o². Determine whether x(t) is a wide sense stationary process or not.If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.16) A stationary random process has an autocorrelation function of Rx (T)=16-e cos 20rt+8 cos10TT. Find the variance of this process.
- 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).5. Let Y,, Y2, ., Yn be independent, exponentially distributed random variables with mean 0/2. Show that the variance of the minimum, Y1) = min(Y,, Y2, ...,n), are E(Y1)) Var(Ya)) = and 2n 02 4n²°3. Consider the MA(2) model y, = ɛ, + 0ɛ,-2. where < 1 and (s,) WN(0, 1). (a) Find the autocovariance functions of (y,). (b) Is this model weakly stationary? (c) Compute the variance of the sample mean (y, + y2 + ys)/3 when 0 0.5.
- A firm's revenue R is stochastically related to the effort exerted by its employee. Effort is a continuous variable. The employee can choose any level of effort e E [0, ). The choice of effort affects revenue so that: E(R|e) = e and Var(R|e) = 1 %3D where E(R|e) and V ar(R|e) denote the expected value and variance, respectively, of rev- enue when the employee exerts effort level e. The employer cannot observe the level of effort exerted by the employee. The employer wants to design a wage contract w based on the revenue and considers only contracts of the form: w-α+ βR and so the employee is guaranteed a payment a and then a bonus payment ßR which de- pends on revenue. The employee is a risk-averse expected utility maximiser. A contract w gives expected utility: Eu(w\e) = E(w\e)-eV ar(w]e) – c(e) where E(wle) and Var(wle) denote the expected value and variance of the contract, re- spectively, conditional on effort e, p is a parameter of risk aversion, and c(e) denotes the disutility of…Example 24: If a random variable X has the moment generating function Мx (t) — 2-t' determine the variance of X.(16) The moment-generating function of the geometric random variable X with parameter p is M(t) = 1-per. Use this to find the mean and variance of X.