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) ?
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- Find the PSD of a stationary random process for which Auto correlation is Rx (t)= 6- e altiThe random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−3/4. Calculate Var(X+Y) with a full explanationLet X and Y have the joint pdf f(x,y)= x+y , 0<=x<=1, 0<=y<=1. Calculate the mean(x) mean (y) variance (x) variance(y)
- Let I be an indicator variable with p(I) = {p , I=1 } {1-p , I=0}. Show that the variance of I is p(1-p)The p.d.f. of a random variable X' is as shown in the figure. The pdf is zero for X 5. Calculate (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X. fx (x) kIf X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.
- The auto correlation function of an a periodic random process is Rxx (T) = exp Find the PSD and average power of the signal. %3D 2 o?X is a discrete uniform (1,3) random variable. When X=x, Y is discrete uniform (1,x). (a) Find the correlation of X and Y (b) Find the mean value of X when Y=2.Prove that the variance of b, V(b) = o²(X'X)−¹
- An investor has found that company1 have an expected return on E(X) = 4% and variance for the return equal V(X) = 0.49. Company 2 has E(Y) = 6% and variance V(Y) = 0.64. The correlation between the companies return is ρ(X,Y) = 0.3. The investor wants to invest p (0<p<1) in company1 and (1-p) in company2. The combined investment have a return: R = pX + (1-p)Y. Let p=0.4 such that R= 0.4X +0.6Y. Find the Expectation and variance of R. how are these results in comparison with X and Y separatley?Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; isThe 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?