Inverse Transform Sampling

Inverse Transform Sampling

Inverse Transform Sampling is a statistical technique used to generate random samples from a specified probability distribution. The method utilizes the cumulative distribution function (CDF) of the distribution, which describes the probability that a random variable takes a value less than or equal to a given point.

The process begins by generating a uniform random number, U, from the interval [0, 1]. This value represents a probability. The next step is to apply the inverse of the CDF, denoted as F^-1(U), to this random number. The result is a sample X drawn from the desired distribution.

For example, if you want to sample from an exponential distribution with rate parameter λ, you would first generate a uniform random number, U. Then, you would compute the inverse CDF (or quantile function) for the exponential distribution, which is X = -ln(1 – U) / λ. This will yield a random sample from the exponential distribution.

Inverse Transform Sampling is particularly useful because it provides a straightforward way to sample from various distributions, making it a popular choice in Monte Carlo simulations and probabilistic models. However, it is important to note that this method may not be efficient for all distributions, especially those without a simple or computable inverse CDF.