Uniform Distribution

• If we want a random variate $X$ uniformly distributed on the interval [a,b], a reasonable guess for generating $X$ is given by
  
  $$X = a + (b-a)R$$
  
  
  where $R$ is uniformly distributed on (0,1).

  If we follow the steps outlined in previous section, we get the same result.

• pdf for X
  
  $$f(x) = \left\{ \begin{array}{ll} \frac{1}{b-a} & a \le x \le b \\ 0 & otherwise \\ \end{array} \right.$$
  
  

• steps:

  Step 1.

  the cdf
  

  $$F(x) = \left\{ \begin{array}{ll} 0 & x < a \\ \frac{x-a}{b-a} & a \le x \le b \\ 1 & x > b \\ \end{array} \right.$$

  
  

  Step 2.

  Set $F(X) = (X - a)/(b - a) = R$

  Step 3.

  Solve for $X$ is terms of $R$ yields
  

  $$X = a + (b-a)R$$

  
  

  which is the same as the earlier guess.

  Step 4.

  Generate $R_i$ as needed, calcualte $X_i$ using the function obtained.