2. A stationary stochastic process has the covariance functions (a) R(T) = e-a|T| (b) R(T) = A + B cos WOT Determine the corresponding spectral density functions.
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- Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such that the initial population at time t=0 is P(0)=P0. Show algebraically that cP(t)P(t)=cP0P0ebt .Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.3. If X and Y are jointly continuous random variables with joint PDF fx.y (r, y) = exp(-y)I0.v)(x)I(0,0) (y) (a) Find the joint MGF of X and Y (b) Find the marginal MGFS of X and Y (c) Identify the marginal distribution of X and Y (d) Find the E(XY) using the joint MGF of X and Y
- A (t) is a random process having mean = 2 and auto correlation function Rxx (7) = 4 [e- 0.2 ld+ 1. Let Y and Z be the random variables obtained by sampling X (t) at t = 2 and t = 4 respectively. Find the variance of the random variable W = Y -Z.The joint PDF of the random variables X and Y is defined as f(x, y) = 25e³"; 0 0 = 0, otherwise %3D (i) (ii) Find the marginal PDFS and X and Y What is the covariance of X and Y?If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.
- 2 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).Which of the following could be the correlation function of a stationary random process? (a) R(t) = exp(-7). (d) R(T) = sin(7). (b) R,(t) exp(-г)и( г). (e) R(7) = cos (r). %3D (c) R (7) = exp(l1). (f) R(7) = sinc(7). %3DConsider 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) ?
- X (t) is a random process having mean = 2 and auto correlation function. Rxx (7) = 4 [e-0.2 ltl + 1. Let Y and Z be the random variables obtained by sampling X (t) att = 2 and t = 4 respectively. Find the variance of the random variable W = Y -Z.(47) Let X z b(8,-) find E(5+6x) and distribution function.3. Let X and Y be two continuous random variables with joint PDF of + 0
