Let X(t) be a Yule process with positive birth parameter ẞ that isobserved at a random time U, where U is uniformly distributed over [0,3). Alsoassume that the Yule process X (t) and the random time U are independent, andX(0) = 1.(i) Find P(X(U) = k) for k ≥ 1.(ii) Find E[X(U)].

Part (i): Finding P(X(U) = k)Step 1: Understand the Problem:We have a Yule process, X(t), with birth…

Answered: Let X(t) be a Yule process with…

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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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).
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Let X(t) be a Yule process with positive birth parameter ẞ that is observed at a random time U, where U is uniformly distributed over [0,3). Also assume that the Yule process X (t) and the random time U are independent, and X(0) = 1. (i) Find P(X(U) = k) for k ≥ 1. (ii) Find E[X(U)].