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”

$$\begin{align} \mathbb{P}[X \in A] &= \int_{\Omega_Y}\mathbb{P}[Y > y \mid X= x] f_Y(y) dy\\ &= \int_{\Omega_Y}\int_{A}f_{X\mid Y}(x\mid y) f_Y(y) dxdy \end{align}$$