Assume that X(t) and Y(t) are two independent standard Brownianmotion processes satisfying X (0) = 0 and Y(0) = 0. Let M(t) = maxoz, X(t) ≤ x) = 1 - (2z-√t2 > 0, x < 2,where denotes the cumulative distribution function of a standard normal.(ii) Find the joint probability density function, fM(t),x(t) (z, x), of M(t) and X(t)for t> 0.[Hint: Apply Reflection Principle for (i).]

Recognizing the ProblemThe maximum value of X(t) up to time t, represented by M(t), is of importance…

Answered: Assume that X(t) and Y(t) are two…

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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Exercise 19. Let X and Y be two random variables with joint probability density function given by fx,y (x, y) = 2e¬(atu) I10,2] (y)/10,00) (Y). Find the joint probability density function of X and X +Y.
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Let X and Y have the joint probability density functionf(x, y) = 5, for 0 < x < y < 1.(i) Find the marginal probability density function of X and Y.(ii) Find the conditional probability density function of X, given Y = y.(iii) Find the conditional mean of X, given Y = y.(iv) Find the conditional variance of X, given Y = y.
The probability density function of a distribution is given by x f(x) = exp(-7). Use differentiation to show that the probability density function has a maximum at x = 0. The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1], q = 1 - p and n is a positive integer. Use the moment generating function to find the mean and variance of the distribution in terms of p, q and n.
Q1) CDF, PDF, Expectation and Variance The cumulative distribution function (CDF) of random variable Y is y 1. a) Find the probability density function, fy(y), of random variable Y. b) Show that f fy (y)dy = 1. c) What is E[Y]? d) What is VAR[Y]?
Let X and Y be independent random variables with joint probability density function fxy(x, y) = 1/3 (x + y), 0 < x <= 2, and 0 < y<= 1, and 0 otherwise. The marginal pdf fx(x) is given by O a. O b. O c. O d. (2 +2X)/3 (2 + 2X)/3 (X+1/2)/3 (X+1/2)/3 0 < X<= 2 0< X<= 1 0 < X<= 1 0
Suppose that X and Y are statistically independent and identically distributed uniform random variables on (0,1). (a) Write down the joint probability density function fx,y(x,y) of X and Y on its support. (b) The expression for the joint probability density function of the transformed random variables U=5X+Y and V=3X+2 Y on its support is: fu,v(u,v) = Au³ (C + D)E Which values of the constants A, B, C, D, E are correct (in the same order as they appear here)? ○ 1,8,1,8,1 ○ none of the answers is correct The joint probability density is constant on its support, hence B-C=0 and D=E=1, and A following from normalization. ○ 1/3, 1, 1.67, 1, -2 O 1/3, 1, 1.67, 1, 2 3, 1, 1.67, 1, -2
Suppose two independent continuous random variables X, Y have exponential distribution with mean 0. a) Write the joint probability density function. b) Express F (2) = P( < z) using integrals and compute the integrals. c) Calculate the PDF of Z = Y/X as f(x) = F' (2) d) Compute the expected value and the variance of Z, if they exist.

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