Examples with functions of Uniform Random Numbers 10/5

Example 1:
The cdf of a uniform random variable is:

$$F_U(u)=\left\{ \begin{array}{llr} 0 &\mbox{ for } & u<0\\ u... ...x{ for } & 0<u<1\\ 1 &\mbox{ for } & u>1\\ \end{array}\right.$$

Thus by computing the derivative we have the density of the uniform random variable to be:

$$f_U(u)=\left\{ \begin{array}{llr} 0 &\mbox{ for } & u<0\\ 1... ...x{ for } & 0<u<1\\ 0 &\mbox{ for } & u>1\\ \end{array}\right.$$

The box shape we already knew!

Example 2:
The square of a random variable:

$$Z=U^2, \mbox{ with } U \; \; Uniform(0,1)$$

We start by computing the cdf by

$$For\; 0<z<1\;\; P(Z \leq z)=P(U^2 <z)=P(U<\sqrt{z})=\sqrt{z}$$

$$F_Z(z)=\left\{ \begin{array}{llr} 0 &\mbox{ for } & z<0\\ \... ...} &\mbox{ for } & 0<z<1\\ 1 &\mbox{ for } & z>1\\ \end{array}$$

We obtain the density by just deriving this cdf:

$$f_Z(z)=\left\{ \begin{array}{llr} 0 &\mbox{ for } & z<0\\ \... ...} &\mbox{ for } & 0<z<1\\ 0 &\mbox{ for } & z>1\\ \end{array}$$

Example 3:
The sum of two uniform random variables:

Z=U₁+U₂

$$For\; z<0\;\; P(Z \leq z)=0$$

$$For\; 0<z<1\;\; P(Z \leq z)=P(U_1+U_2 <z)=\frac{z^2}{2}$$

$$For\; 1<z<2\;\; P(Z \leq z)=P(U_1+U_2 <z)=1-\frac{(2-z)^2}{2}$$

We obtain the density by just deriving this cdf:

$$f_Z(z)=\left\{ \begin{array}{llr} 0 &\mbox{ for } & z<0\\ z... ...) &\mbox{ for } & 1<z<2\\ 0 &\mbox{ for } & z>2\\ \end{array}$$

This the triangle shaped density that we found by simulation.