CS(14) Transformation Methods: Mixtures

Mixtures involve combining multiple distributions to model scenarios where the data arises from heterogeneous sources or subpopulations. Mixtures are broadly classified into discrete mixtures, where the combination involves distinct probability distributions, and continuous mixtures, where a continuous random variable governs the combination process. These methods are particularly useful for representing complex distributions that cannot be adequately modeled by a single probability law.

Discrete Mixtures

A discrete mixture is formed by combining general distributions using weighted probabilities. Consider a random variable X with a distribution represented as:

Each 𝜽i is always >0 and represents the probability of selecting the i-th distribution, and its sum is 1.

For example, suppose we want to generate a random variable that is a mixture of two normal distributions N(0,1) and N(4, 1), with equal weights 𝜽1= 0.5 and 𝜽2=0.5. The procedure involves:

→ Randomly selecting one of the distributions with the probabilities 𝜽1 and 𝜽2, and
→ Generating a sample from the chosen distribution.