Q. 10.7 - A First Course in Probability

Question

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) … x} = xn 

(c) Use part (b) to give a second method of simulating a random variable having distribution F.

Step-by-Step Solution

Answer

(a) University of the unform.

(b) The statement is proved below.

(c) The distribution will be followed with the CDF.

Part (a) Step 1: Given Information

We have to find a method for simulating a random variable having distribution F that uses only a single random number.

Part(a) Step 2: Explanation

Consider$F\left(x\right)=x^n$ for $0<x<1$, which implies$F^{-1}\left(y\right)=y^{\frac{1}{n}}$ . So, taking a random number $U\in \left(0,1\right)$ and generating $X=U^{\frac{1}{n}}$. From the universality of the Uniform, we have$X$follows required distribution.

Part (b) Step 1: Given Information

We need to show that

$P\left\{max(U_1,...,U_n)\le x\right\}=x^n$.

Part (b) Step 2: Simplify

By the independence, we have

$\begin{array}{rr}P\left(m \left(U_1,\dots ,U_n\right)\le x\right)& =P\left(U_1\le x,\dots ,U_n\le x\right)\\ & =P\left(U_1\le x\right)\dots P\left(U_n\le x\right)\\ & =x^n\end{array}$

Part (c) Step 1: Given Information

We have to use part (b) to give a second method of simulating a random variable having distribution F.

Part (c) Step 2: Explanation

Generating independent Uniforms on $\left(0,1\right)$ , call them $U_1,\dots ,U_n$ and considering its maximum - call it $X$.

From the part (b), we have$X=\text{max}\left(U_1,\dots ,U_n\right.$ follows the distribution with CDF $F\left(x\right)=x^n$.