Käänteisvaihtomenetelmä

Käänteisvaihtomenetelmä, also known as the inverse transform sampling method, is a technique used in probability and statistics to generate random numbers from a specified probability distribution. It is particularly useful when direct sampling from the desired distribution is difficult or computationally expensive. The core principle of the method is to use the inverse of the cumulative distribution function (CDF) of the target distribution.

To apply the käänteisvaihtomenetelmä, one first needs to determine the CDF of the desired probability distribution. Let this CDF be denoted by F(x). The next step is to find the inverse function of F(x), denoted as F⁻¹(u). This inverse function takes a probability value between 0 and 1 as input and outputs a value from the random variable's domain.

Once the inverse CDF is obtained, the method proceeds by generating a random number U from a uniform distribution between 0 and 1 (U ~ Uniform(0, 1)). This random number U is then passed through the inverse CDF function: X = F⁻¹(U). The resulting value X is a random sample drawn from the target distribution F(x).

The käänteisvaihtomenetelmä is widely applicable and forms the basis for many simulation techniques in various fields, including finance, physics, and machine learning. Its effectiveness relies on the ability to analytically or numerically derive the inverse of the cumulative distribution function, which can sometimes be a complex mathematical task.