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Chapter 10: Q. 10.4 (page 430)

Present a method for simulating a random variable having distribution function

F(x)=0x鈮�-312+x6-3<x<012+x2320<x鈮�41x>4

Short Answer

Expert verified

The universality of uniform is used to obtain the required method.

Step by step solution

01

Given Information

We have given the function

F(x)=0x鈮�-312+x6-3<x<012+x2320<x鈮�41x>4

02

Simplify

Finding the value F-1. fory鈭�(0,1/2)we have

y=12+x6

x=6y-3

and fory鈭�(1/2,1) we have

y=12+x232

x=42y-1

So, the method is as follows. choose a random number (call it y) from the interval (0,1). If it is less than 12, declare x = 6y - 3. And if it is greater than 12, declare x=42y-1. from the universality of the uniform, we have that x follows the distribution of X.

y-12+x6x=6y-3y=12+x6x=6y-3

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Most popular questions from this chapter

(a) Verify that the minimum of (4.1) occurs when a is as given by (4.2).

(b) Verify that the minimum of (4.1) is given by (4.3).

Let F be the distribution function

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(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.

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