Show that the random process X(t) = A cos (@nt + 0) is wide-sense stationary if it is assumed that A and o, are constants and O is a uniformly distributed random variable on the interval (0, 2n). %3D
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- Show that the random process X(t) = A cos (@nt + 0) is wide-sense stationary if it is assumed that A and wn are constants and 0 is a uniformly distributed random yariable on the interval (0, 2n).A random process is described by X(t) = A , where A is a continuous random variable uni formly distributed on (0,1). Show that X(1) is stationary process.b) A random variable X follows an exponential distribution, X~Exp(0), with parameter 0 > 0. Find the cumulative distribution function (CDF) of X. i) ii) Show that the moment generating function (MGF) of X is M(t) = iii) Use the MGF in (ii) to find the mean and variance of X.
- Suppose the random variable X~ Beta(a, 3), namely its PDF fx(x) = I(a + 3) r(a)(3) -ra-1(1−z)3-1, 0 < x < 1, Beta(a +3, y), namely its PDF fy (y) = I(a +8+y) r(a + 3)(y) ¹(1 − y)*-¹, 0 < y < 1, and X and Y are independent. Define U = XY, V = X. Find the joint PDF fuv(u, v).Assume that X and Y are independent random variables where X has a pdf given by fx(x) = 2aI(0,1)(x) and Y has a pdf given by fy(y) = 2(1– y)I(0,1)(y). Find the distribution of X + Y.Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)
- A projectile is launched at an angle theta with respect to the surface with velocity v0 (deterministic). If the angle of inclination is a uniform random variable in [0, pi/2 ], calculate the distribution function of the variable R defined as the point of impact of the projectile on the ground, measured from the origin. Also calculate your expected valueSuppose that X and Y are continuous random variables with joint pdf f (x, y) = e-**) 0Let P be a random variable having a uniform distribution with minimum 0 and maximum 3 i.e. P ~ Uniform(0, 3). Let Q = log (P/(3-P)). Find E[P]. You areexpected to solve this problem without using Method of Transformation.Two random processes are defined as X (t) = A. cos (ot + 0); Y (() = A sin (@t + 0), where A and o are constants and e is a uniform random variable over (0, 2n). Verify that Rxy (T) = Ryx (- ).The two random processes X(t) and Y(t) are defined as X(t) A cos (o, t) + B sin (@, t) Y(t) = B cos (oo )-A sin (@ot) where A and B are random variables, on is a constant. Show that, X(t) and Y(t) are jointly wide-sense stationary. Assume that A and B are uncorrelated, zero-mean random variables with same variance irrespective of their density functions.Let x(t) be a wide sense stationary random process with auto correction function R(t) = elt %3D XX Where a > 0 is a constant. We assume x(t) "amplitude modulates" a "carrier" cos(@,t + 0), where o, is a constant and 0 is a random variable uniform on (-T, T) that is statistically independent of x(t). Determine the auto correlation function of y(t). GESEE MORE QUESTIONS