CS(12) Inverse Transformation Method for Generating Random Variables

The Inverse Transformation method is a fundamental technique for generating random variables from any probability distribution, provided the cumulative distribution function (CDF) of the desired distribution is known and invertible. This method leverages the uniform distribution as a starting point and applies a mathematical transformation to produce random variables following the target distribution.

To understand the inverse transformation method, we must first consider the cumulative distribution function (CDF), F(x), which describes the probability that a random variable X takes a value less than or equal to x.

Mathematically, F(x) = P (X ≤ x). The CDF has the property that F(x) increases monotonically from 0 to 1 as x moves from negative infinity to positive infinity. If we have a uniform random variable U ∼ U (0,1), the key idea of the inverse transformation method is to map U to X using the inverse CDF, denoted 1/F¹(u).

The transformation is given by X = 1/F¹(u). This ensures that X follows the distribution defined by F(x). The reasoning behind this lies in the property that P(1/F¹(u) ≤ x) = F(x), which matches the desired distribution.

Let us break this down with an example:
Let’s generate random variables from an exponential distribution. The…