In this exercise we examine a useful technique for generating a set of samples having a given pdf.

(a) First, prove the following theorem: If X is a con tinuous random variable with cdf F_X(x), the random variable

Y = F_x(X)

is a uniformly distributed random variable in the interval (0, 1).

(b) Using this theorem, design a random number generator to generate a sequence of exponentially distributed random variables having the pdf

f_X(x) = αe^-αxu(x)

where u(x) is the unit step. Plot histograms of the random numbers generated to check the validity of the random number generator you designed.