The definition and interpretation of conditional distributions for continuous random variables are similar to those for discrete variables.

Definition

The conditional distribution of \(Y\) given \(X=x\) is the distribution with probability density function

\[ f_{Y \mid X=x}(y) \;\;=\;\; \frac{f(x,y)}{f_X(x)} \]

Its shape is that of a slice through the joint pdf at \(X=x\), but it is scaled to have unit area by dividing by the area of the slice.

Example

The random variables \(X\) and \(Y\) have joint probability density function

\[ f(x,y) \;=\; \begin{cases} x+y & \quad\text{if }0 \lt x \lt 1 \text{ and }0 \lt y \lt 1 \\ 0 & \quad\text{otherwise} \end{cases} \]

What is the conditional pdf of \(Y\), given that \(X = x\)?

What is the probability that \(Y\) is more than 0.5, given that \(X\) is 0.7?

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