Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Q. 10.5 (page 432)

Let X and Y be independent exponential random variables with mean 1.

(a) Explain how we could use simulation to estimate E[eXY].

(b) Show how to improve the estimation approach in part (a) by using a control variate.

Short Answer

Expert verified

(a) Estimator M=1n∑i=1neXiYi
(b) Redefine estimator Mˆ=1n∑i=1neXiYi+cXiYi−1and it is defined below.

Step by step solution

01

Part (a) Step 1: Given Information

We need to explain the simulation to estimateEeXY.

02

Part (a) Step 2: Simplify

Considering to define X1,…,Xn~Xand Y1,…,Yn~Y.

Now, consider the statistic

M=1ni=1neXiYi

It is simple to show that EM=EeXYso we have provided a good method of estimation of the mean.

03

Part (b) Step 1: Given Information

We need to give the solution to improve the estimation approach in part (a) by using a control variate.

04

Part (b) Step 2: Simplify

Consider

Mˆ=1ni=1neXiYi+cXiYi−1

Serve that it also holds

EMˆ=1ni=1nEeXiYi+cXiYi−1=EeXY

as we haveEcXiYi−1=0. Let's minimize the variance. We have that

VarMˆ=1n2i=1nVareXiYi+cXiYi−1

Keeping that

VareXiYi+cXiYi−1=VareXiYi+c2VarXiYi−1+cCoveXiYi,XiYi−1

So, we have seen that the Variance is minimized when

c=−CoveXiYi,XiYi−1VarXiYi−1

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose we have a method for simulating random variables from the distributions F1 and F2. Explain how to simulate from the distribution

F(x) = pF1(x) + (1 − p)F2(x) 0 < p < 1

Give a method for simulating from

F(x)13(1-e-3x)+23x0<x≤113(1-e-3x)+23x>1

The random variable X has probability density function

f(x) = Cex 0 < x < 1

(a) Find the value of the constant C.

(b) Give a method for simulating such a random variable.

The following algorithm will generate a random permutation of the elements 1, 2, ... , n. It is somewhat faster than the one presented in Example 1a but is such that no position is fixed until the algorithm ends. In this algorithm, P(i) can be interpreted as the element in position i. Step 1. Set k = 1. Step 2. Set P(1) = 1. Step 3. If k = n, stop. Otherwise, let k = k + 1. Step 4. Generate a random number U and let P(k) = P([kU] + 1) P([kU] + 1) = k Go to step 3. (a) Explain in words what the algorithm is doing. (b) Show that at iteration k—that is, when the value of P(k) is initially set—P(1),P(2), ... ,P(k) is a random permutation of 1, 2, ... , k.

Let (X, Y) be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thus

f(x,y)=1π0≤x2+y2≤1

Let R = (X2 + Y2)1/2 and = tan−1(Y/X) denote

the polar coordinates of (X, Y). Show that R and are

independent, with R2 being uniform on (0, 1) and being

uniform on (0, 2π).

In Example 4a, we showed that

E[(1 − V2) 1/2] = E[(1 − U2) 1/2] = π/4

when V is uniform (−1, 1) and U is uniform (0, 1). Now show that

Var[(1 − V2) 1/2] = Var[(1 − U2) 1/2]

and find their common value.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free