A random variable is a function that maps probabilistic outcomes to quantities of interest . In layman's terms, is a variable that can take on different states at random.

Let be the target space containing all such . A random variable can be expressed as where is the outcome space, defined in context of a 🎲 Probability Distribution.

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For example, in the case of tossing two coins, . If we're interested in the number of heads, and , , and .

Transformations#

Sometimes, we're given random variable and a transformation . Given the distribution for , our task is to find the distribution for . The two core methods are detailed below.

Distribution Function#

This technique first finds the cdf of , then differentiates it to get the pdf.

1. Find by substituting into .
2. Differentiate to get the pdf.

Change of Variables#

The change of variables technique uses similar steps from the distribution function to generalize one final formula,