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Chapter 10: Q. 10.12 (page 431)

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

Short Answer

Expert verified

The answer of the question is鈭�01k(u)du=E(k(U))=1n鈭�i=1nk(Ui)

Step by step solution

01

Given Information

We need to explain the use of random numbers to approximate鈭�01k(x)dx

02

Simplify

Consider U~Unif(0,1).Using the theorem about the expectation of the function of the random variable, we have that

E(k(u))=鈭�鈩�k(u)fU(u)du=鈭�01k(u)du

On the other hand, we can estimate the mean of random variable with the mean of the sample. So, taking the independent variable U1,...,Un~Uand considering its functional values localid="1648210961949" k(U1),...,k(Un).Therefore, we can estimate

E(k(U))=1n鈭�k(Uii=1n)

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Most popular questions from this chapter

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.

Give an efficient algorithm to simulate the value of a random variable with probability mass function

p1 = .15 p2 = .2 p3 = .35 p4 = .30

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

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.

In Example 2c we simulated the absolute value of a unit normal by using the rejection procedure on exponential random variables with rate 1. This raises the question of whether we could obtain a more efficient algorithm by using a different exponential density鈥攖hat is, we could use the density g(x) = 位e鈭捨粁. Show that the mean number of iterations needed in the rejection scheme is minimized when 位 = 1.
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