[SOLVED] EE 381 - Project 4

Simulating Random Variables

Inverse Transformation Method

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:

𝐹_𝑇(𝑡) = 1 − 𝑒^−𝜆𝑡 for 𝑡 ≥ 0  and 𝑓_𝑇(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 𝑓_𝑈(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 𝐹_𝑇(𝑡) is strictly increasing on the interval [0, ∞).  Let 0 < 𝑢 < 1 and observe that there is a unique value of 𝑡 such that 𝐹_𝑇(𝑡) = 𝑢.  Thus 𝐹_𝑇⁻¹(𝑢) for 0 < 𝑢 < 1 is well defined.  In this case 𝐹_𝑇(𝑡) = 1 − 𝑒^−𝜆𝑡 = 𝑢 if and only if.  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.