Conditional PMF and PDF
- Conditional PMF
\[ \begin{align} p_{X\mid Y}(x\mid y) &= \mathbb{P}[X = x \mid Y = y] \\ &=\frac{\mathbb{P}[X=x \cap Y=y]}{\mathbb{P}[Y=y]}\\ &= \frac{p_{X,Y}(x,y)}{p_Y(y)} \end{align} \]
- Useful result: if you have conditional distribution and need marginal, to for example compute probability of event in marginal:
\[ \begin{align} \mathbb{P}[X \in A] &= \sum_{x \in A}\sum_{\Omega_Y}p_{X, Y}(x,y) \\ &= \sum_{x \in A}\sum_{\Omega_Y}p_{X\mid Y}(x\mid y) p_Y(y) \end{align} \] - Can also define conditional CDF:
\[ F_{X\mid Y}(x \mid y) = \sum_{x'\leq x}p_{X \mid Y}(x'\mid y) \]
- For continuous case the conditional pdf is defined:
\[ f_{X\mid Y}(x\mid y) = \frac{f_{X,Y}(x,y)}{f_Y(y)} \]
- Conditional CDF
\[ F_{X\mid Y}(x\mid y) = \frac{\int_{-\infty}^{x}f_{X,Y}(x',y)dx'}{f_Y(y)} \]
Book uses conditional CDF to justfy the PDF.
- Compute probability of event in margin using conditional pdfs”
\[ \begin{align} \mathbb{P}[X \in A] &= \int_{\Omega_Y}\mathbb{P}[Y > y \mid X= x] f_Y(y) dy\\ &= \int_{\Omega_Y}\int_{A}f_{X\mid Y}(x\mid y) f_Y(y) dxdy \end{align} \]