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Conditional probability density function

by Marco Taboga, PhD

The probability distribution of a continuous random variable can be characterized by its probability density function. When the probability distribution of the random variable is updated, by taking into account some information that gives rise to a conditional probability distribution, then such a distribution can be characterized by a conditional probability density function.

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The following is a formal definition.

Definition Let X and Y be two absolutely continuous random variables. The conditional probability density function of X given Y=y is a function [eq1] such that[eq2]for any interval [eq3].

In the definition above the quantity [eq4]is the conditional probability that X will belong to the interval $\left[ a,b\right] $, given that Y=y.

More details

More details about the conditional probability density function can be found in the lecture entitled Conditional probability distributions.

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