Question

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

(a) The value of the constant is$C=\frac{1}{e-1}$.

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

Step-by-step solution

Part (a) Step 1: Given Information

We need to find the value of the constant$C$.

Part (a) Step 2: Simplify

There have to be ${\int }_0^{1}f\left(x\right)dx=1.$Hence

$1={\int }_0^{1}C{e}^{x}dx=C{e}^x|\underset{0}{\overset{1}{}}=C(e-1)$

which implies

$C=\frac{1}{e-1}$

Part (b) Step 1: Given Information

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

Part (b) Step 2: Simplify

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

localid="1651486522729" $F\left(x\right)={\int }_0^x\frac{1}{e-1}{e}^{S}ds=\frac{1}{e-1}{e}^S{|}_0^x=\frac{e^x-1}{e-1}$

Now, for $F^{-1}$

$y=\frac{e^x-1}{e-1}\Rightarrow y(e-1)+1=e^x\Rightarrow x\log \left(y\right(e-1)+1)$

so $F^{-1}\left(u\right)=\log \left(u\right(e-1)+1)$.The method is as follows. Generate $U$uniformly from $(0,1)$. Calculate $F^{-1}\left(U\right)$and declare that it is equal to $X$. By the universality of the uniform, we have that $X$has required PDF.