(a) Let (X{)t>o be a stochastic process defined byfr dW,,t > 0,where W = (W,)r>o_is a Wiener martingale and f € H. Provert^sCov(X3, X4) = E(X,X;) = E f; dr,for any s,t e [0, 0),S-where s At = min(s,t).

(a)In this problem, stochastic process is given which is defined as the integral of a function f…

Answered: (a) Let (Xt)t≥o be a stochastic process…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Let X be a continuous random variable whose moment generating function is 4x(t) = (5) ³. Find Var (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.
Suppose that X and Y are continuous random variables with joint pdf f (x, y) = e-**) 0
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.
Find P(X=4) if X has a Poisson distribution such that 3P(X=1)=P(X=2)
2) If the random variable (x) uniformly distributed on the interval (-2,4), then the moment generating function Mx(t) = e4t а) -e-2t e4t. b) -e-2t -6t 6t e 4t -e2t e4t d) +e-2t 6t 6t
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).
2 Let X (t) be a random process with mean 3 and auto correlation R(t, t2) = 9 + 4 e-0.2 ,-t, Determine the mean, variance and covariance of the random variables Z =X (5) and wX (8).
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)
Q3/(a) Let X = b(n,) and E(x)= E(16-7x) find n. (b)Let X be a random variable with p.d.f, ƒ(x)= c 3 x=1,2,3,.., zero elsewhere. Find the constant c.
Example 18.2 For the probability density of a system of random variables (X, Y): f(x, y) = 0.5 sin (x + y) (0 < x < 7,0 << 7), 2' determine (a) the distribution function of the system, (b) the expectations of X and Y, (c) the covariance matrix.
2. Let U1, U2,... be independent random variables, each with continuous distribution that is uniform in the interval [-3, 3]. Let S, = U ++ Un. (a) Compute the moment generating function of U1. (b) Compute the moment generating function of S,. (c) Determine E(S) and E(S) (by any method).

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(a) Let (Xt)t≥o be a stochastic process defined by f fr dWr, t≥0, D where W = (Wr)rzo is a Wiener martingale and fE H. Prove rt^s € Cov(X§, Xt) = E(X¸Xt) = E [^³ f² dr, for any s,t = [0, ∞), where s ^ t = min(s, t).