2023-11-15
What do we know about conditional probabilities for events and discrete RVs?
For events:
\[P(A | B) = \dfrac{P(A \cap B)}{P(B)}\]
For discrete RVs: \[p_{X|Y}(x|y) = P(X=x|Y=y) = \dfrac{p_{X,Y}(x,y)}{p_Y(y)}\]
What does it mean for conditional densities of continuous RVs?
For continuous RVs:
Example 1.1
Let \(f_{X,Y}(x,y)= 5 e^{-x-3y}\), for \(0 < y < \frac{x}{2}\).
Definition: Conditional density
The conditional density of a r.v. \(X\) given \(Y=y\), is \[f_{X|Y}(x|y)= \frac{f_{X,Y}(x,y)}{f_Y(y)},\] for \(f_Y(y)> 0\)
Remarks
For a fixed value of \(Y=y\), the conditional density \(f_{X|Y}(x|y)\) is an actual pdf, meaning
\(f_{X|Y}(x|y)\geq 0\) for all \(x\) and \(y\), and
\(\displaystyle\int_{-\infty}^{\infty} f_{X|Y}(x|y)dx =1\).
Example 1.1
Let \(f_{X,Y}(x,y)= 5 e^{-x-3y}\), for \(0 < y < \frac{x}{2}\).
Example 1.2
Let \(f_{X,Y}(x,y)= 5 e^{-x-3y}\), for \(0 < y < \frac{x}{2}\).
Example 2
Randomly choose a point \(X\) from the interval \([0,1]\), and given \(X=x\), randomly choose a point \(Y\) from \([0,x]\). Find \(\mathbb{P}(0 < Y < \frac14)\).
Question What is \(f_{X|Y}(x|y)\) if \(X\) and \(Y\) are independent?
\[f_{X|Y}(x|y) = \dfrac{f_{X,Y}(x,y)}{f_y(y)} = \dfrac{f_{X}(x)f_y(y)}{f_y(y)} = f_{X}(x)\]
Chapter 27 Slides