- Consider the continuous-amplitude, continuous-time random process X(t), te R defined by: X(t) = Y +tY₂ where Y, and Y are independent Gaussian random variables, each having zero mean and variance σ2. Find the one- dimensional and two-dimensional density functions for this random process.
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- (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.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) · dtA (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.
- Q. 3 Let continuous random variables X and Y be independent identically distributed random variables with the following respective pdfs: fx (x) = 2e-2x; x > 0 and fy (y) = 2e-2y; y > 0 Define a new random variable Z = X + Y. Derive moment %3D generating function of Z, i.e. Mz(t).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) ?If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.
- Q. 3 Let continuous random variables X and Y be independent identically distributed random variables with the following respective pdfs: fx(x) = 2e-2x;x 2 0 and fy (y) = 2e-2y;y > 0 Define a new random variable Z = X + Y. Derive moment generating function of Z, i.e. Mz(t).Let the random process X(t) be defined as X(t) = A + Bt, where A and B are independent random variables each that is uniformly distributed on (-1,1). Calculate the following: a) Mean mx(t) b) Correlation Rx(t1,t2).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.
- Let X be a (continuous) uniform random variable on the interval [0,1] and Y be an exponential random variable with parameter lambda. Let X and Y be independent. What is the PDF of Z = X + Y.Let the random variable X have the moment generating function M(t) = e³t 1-t² = -1 < t < 1 What are the mean and the variance of X, respectively?Let Xand Y be two continuous random variables with joint probability density [3x function given by: f(x.y)%D 0sysxsl elsewhere with E(X) = ECX)- EC) - EC*)= ;and E(XY) = 10 3 E(Y*) = - and E(XY) =; %3D Then the value of the variance of 2X+Y is: O 3/80 O 91/320 43/320 7/20