Inverse Transformation Method

Simulating Random Variables
Inverse Transformation Method
EE 381 - Project 4, 5 Points

Introduction: There are a variety of ways of simulating random variables, (rv). In this project
we will explore one of the common methods of simulation. This is the inverse transformation
method. Further, we need a context in which to discuss this method. Consequently, we will
introduce a rv and apply the method to it.
Exponential rv, 𝑇: This rv can be used to model the reliability of an apparatus. If the apparatus
has been in use for any number of hours, it is as good as a new apparatus of the same kind in
regards to the amount of time remaining until the item fails. The cumulative distribution
function (CDF) and the probability density function (pdf) are:
𝐹𝑇
(𝑡) = 1 − 𝑒
−𝜆𝑡 for 𝑡 ≥ 0 and 𝑓𝑇
(𝑡) = 𝜆𝑒
−𝜆𝑡
 for 𝑡 ≥ 0.
The inverse transformation method: We will be using a linear congruential pseudorandom
number generator to provide us with a random variable uniformly distributed between zero and
one. This pseudorandom number generator is provided in computer languages. We will
characterize this as: The random variable 𝑈 such that 𝑈 is uniform on the interval [0,1) or
equivalently 𝑓𝑈(𝑢) = 1 for 0 ≤ 𝑢 < 1.
Then the method is based on the argument: For the CDF function 𝐹 if we define the rv 𝑇 by 𝑇 =
𝐹
−1
(𝑈) then the rv 𝑇 has CDF 𝐹.
The application of the inverse transformation method to the exponential distribution. (In doing
Monte Carlo studies it is sometimes necessary to generate a series of exponential RV’s.) Let 𝑈
be a uniform rv on the interval [0,1). Find a transformation such that it possess an exponential
distribution with mean 1⁄𝜆.
The CDF 𝐹𝑇
(𝑡) is strictly increasing on the interval [0, ∞). Let 0 < 𝑢 < 1 and observe that
there is a unique value of 𝑡 such that 𝐹𝑇
(𝑡) = 𝑢. Thus 𝐹𝑇
−1
(𝑢) for 0 < 𝑢 < 1 is well defined. In
this case 𝐹𝑇
(𝑡) = 1 − 𝑒
−𝜆𝑡 = 𝑢 if and only if 𝑡 = −
1
𝜆
ln(1−𝑢)=𝐹𝑇
−1
(𝑢). So, consequently given a
list of random numbers that are uniformly distributed a list of random numbers that are
exponentially distributed can be determined using the derived transformation.
Deliverables: Write a program in Python that simulates an exponential rv using the inverse
transformation method. In addition to the code the plot(s) of the input and output are wanted.
References
Mathematical Statistic with Applications, 5th Ed. By Wackerly, Mendenhall, and Scheaffer 1995
Introduction to Probability Models, 5th Ed. By S. Ross 1993