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Chapter 10: Q. 10.1 (page 431)

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.

Short Answer

Expert verified

(a) The value of the constant isC=1e-1.

(b) Inverse transformation method is used for simulating such a random variable.

Step by step solution

01

Part (a) Step 1: Given Information

We need to find the value of the constantC.

02

Part (a) Step 2: Simplify

There have to be 鈭�01f(x)dx=1.Hence

1=鈭�01Cexdx=Cex|01=C(e-1)

which implies

C=1e-1

03

Part (b) Step 1: Given Information

We need to find a method for simulating given random variable.

04

Part (b) Step 2: Simplify

Using the inverse transformation method. Let's find CDF firstly. For 0<x<1we have

localid="1651486522729" F(x)=鈭�0x1e-1eSds=1e-1eS|0x=ex-1e-1

Now, for F-1

y=ex-1e-1鈬�y(e-1)+1=ex鈬�xlog(y(e-1)+1)

so F-1(u)=log(u(e-1)+1).The method is as follows. Generate Uuniformly from (0,1). Calculate F-1(U)and declare that it is equal to X. By the universality of the uniform, we have that Xhas required PDF.

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