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Chapter 10: Q. 10.2 (page 430)

Develop a technique for simulating a random variable having density function

f(x)e2x-鈭�<x<0e-2x0<x<鈭�

Short Answer

Expert verified

The information for the function will get by using the universality of uniform distribution.

Step by step solution

01

Given Information

We have given a function

f(x)e2x-鈭�<x<0e-2x0<x<鈭�.

02

Simplify

We have

F(x)=鈭�-鈭�xe2sds=e2x2

and for x>0we have

F(x)=鈭�-鈭�0e2sds+鈭�0xe-2sds=1-e-2x2

Calculating the valueF-1. For Y鈭�0,1/2we have

Y=e2x2鈬�x=12log(2Y)

and for Y鈭�1/2,1we have that

Y=1-e-2x2鈬�x=-12log(2-2Y)

Choose random number (call it Y) from the interval 0,1. if it is less than12, declarex=12log(2Y).If it is greater than 12, declare x=-12log(2-2Y). From the universality of the Uniform, we have xfollows the distribution of X.

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Most popular questions from this chapter

Explain how you could use random numbers to approximate 鈭�01k(x)dx, where k(x) is an arbitrary function.

Use the rejection method with g(x) = 1, 0 < x < 1, to determine an algorithm for simulating a random variable having density function

f(x)60x3(1-x)20<x<10otherwise

Let F be the distribution function

F(x) = xn 0 < x < 1

(a) Give a method for simulating a random variable having distribution F that uses only a single random number.

(b) Let U1, ... , Un be independent random numbers. Show that

P{max(U1, ... , Un) 鈥� x} = xn

(c) Use part (b) to give a second method of simulating a random variable having distribution F.

Let X be a random variable on (0, 1) whose density is f(x). Show that we can estimate # 1 0 g(x) dx by simulating X and then taking g(X)/f(X) as our estimate. This method, called importance sampling, tries to choose f similar in shape to g, so that g(X)/f(X) has a small variance.

In Example 4a, we showed that

E[(1 鈭� V2) 1/2] = E[(1 鈭� U2) 1/2] = 蟺/4

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

Var[(1 鈭� V2) 1/2] = Var[(1 鈭� U2) 1/2]

and find their common value.
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