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Chapter 10: Q. 10.11 (page 431)

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

Short Answer

Expert verified

The algorithm is generateY~g(which is unform) and take random numberU鈭�(0,1).

Step by step solution

01

Given Information

We have given the density function

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

02

Simplify

Finding the upper bound of fon the interval (0,1). Using the differentiation, we have

f1(x)=180x2(1-x)2-120x3(1-x)=0

which implies the equality

3(1-x)=2x鈬�x=35

So, the maximum value of fis assumed to be in point x=35and it is equal to localid="1648210382401" f35鈮�2.0736:=c. So, the algorithm is as follows: generate Y~g(which is uniform) and take random number U鈭�(0,1).Consider if

U鈮�f(Y)cg(Y)

and in that case declare X=Y.Otherwise, go to the step 1again.

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Most popular questions from this chapter

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

f(x,y)=1蟺0鈮�x2+y2鈮�1

Let R = (X2 + Y2)1/2 and = tan鈭�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, 2蟺).

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 鈭� p)F2(x) 0 < p < 1

Give a method for simulating from

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