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 . 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.
