(8) If a random process X(t) with zero mean has the DC component A as Y(t) = A+X(t), then Ryy (T) = A² + Ryx (T)
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- 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 scientist has measured quantities y, x1,and x,. She believes that y is related to x, and x2 through the equation y = ae+Px2 8, where & is a random error that is always positive. Find a transformation of the data that will enable her to use a linear model to estimate B, and B,.X(t) and Y(t) are two random processes with cross correlation, AB [sin(@,t)+ cos(@,(2t + T))] Ryy(t, t + t) = 2 Where A, B and @, are costants. Find the cross power spectrum.
- A random process x(t) is applied to a network with impulse response h(t) u(t) 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. Rxx(t) = u(t) T exp (– bt) (a) Find the auto correlation of y(t) ? (b) What is the average power in y(t) ?Find the variance stabilizing transformation g(y) for a response variable y, if E(y) = µ_and_V(y) = √P.7. Suppose that the random variable X have the pdf f (x) = e-a*/2, x > 0 and 0 otherwise. 2T (a) Find E(X) and Var(X). (b) Find the transformation g(X)= Y and values of a and B such that Y ~ I(a, B).
- Q1: (): Let Y, ., Y, be order statistics of a random sample of size 4 from Exponential (10). Find g(yı), g(y1, Y3)and P(Y, > 100). V4-x2 Q2: (): Let f(x) = ,0 sx< 2, zero elsewhere. Let Y = 1 - Find g(y) by transformation method.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?A WSS process X(t) 3 Sxx (w) = 50r d(w) + 1+ with a mean value 5 and power spectrum is applied to a network with impulse response (0) 2 h(t) = 4 e4|t)| i) Find H(w) of the network.
- A WSS process X(t) 3 Sxx (w) = 507 d(w) + 1+ with a mean value 5 and power spectrum is applied to a network with impulse response (0) 2 h(t) = 4 e4|t| Find H() of the network.Consider a logistic regression system with two features x1 and x2. Suppose 00 = 5, 01 = 0, 02= 0, 03= -5, 04= -1, draw the decision boundary of hø(x) = g(0+ 01 x1+ 02 X2 + 03 xı²+ 04 x2?).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 } = ?
