Show that the random process X(t) = A cos (@n t + 0) is wide-sense stationary it is assumed that A and o, are constants and 0 is a uniformly distributed randor variable on the interval (0 2r)
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- 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.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.Let X be a continuous random variable whose moment generating function is 4x(t) = (5) ³. Find Var (X)
- Let X and Y be two jointly continuous random variables. Let Z = X² + Y². Show that F2(=) = L Fxr(V=- yř 19) – Fxy(-v/= = y"l»)] fv (w) dy .Suppose that Z₁, Z2, ..., Zn are statistically independent random variables. Define Y as the sum of squares of these random variables: n Y=)Z (n>2) i=1 (a) Express the moment generating function My(t) of the random variable Y in terms of moment generating functions involving the random variables Z, i = 1, ..., .., n. (b) Determine My(t) for the special case that Z₁ ~ N(0, 1). (c) For the above special case, calculate E[Y] by using the moment generating function.Let 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.
- Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of YShow 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).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).
- 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)Show that the random process X(t) = A cos (@n t + 0) is wide-sense stationary if it is assumed that A and wo are constants and 0 is a uniformly distributed random %3D variable on the interval (0, 2n).Let X1, X2, X3 be independent Exp()-distributed random vairables. Determine the pdf of X(1)X(3)