05 - Conditional Densities

Conditional Probability

Let $A$ and $B$ be events with $\mathbb{P}(A)>0$ then

Conditional Probability of $B$ given $A$ is

$$\mathbb{P}(B\mid A):=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(A)}$$

Conditional Distributions

Let $A$ be an event with $\mathbb{P}(A)>0$
Let $X$ be a random variable then

$$F_{X\mid A}(x):=\mathbb{P}(X\le x\mid A)=\frac{\mathbb{P}(\{X\le x\}\cap A)}{\mathbb{P}(A)}$$

Conditional Probability Mass Function

Let $A$ be an event with $\mathbb{P}(A)>0$
Let $X$ be a discrete random variable then

$$p_{X\mid A}(x)=\mathbb{P}(X=x\mid A)$$

Conditional Density Function

Let $A$ be an event with $\mathbb{P}(A)>0$
Let $X$ be a continuous random variable and for set $C$ then

$$\mathbb{P}(X\in C\mid A)={\int }_{x\in C}{f}_{X\mid A}(x)dx$$

Conditional Expectation

Let $A$ be an event with $\mathbb{P}(A)>0$
Let $X$ be a random variable

Conditional Expectation of $X$ given $A$ is defined by

1. Discrete

$$\mathbb{E}[X\mid A]=\sum _{x}x{p}_{X\mid A}(x)$$

2. Continuous

$$\mathbb{E}[X\mid A]=\int x{f}_{X\mid A}(x)dx$$

Conditional Density Function Given

Let $X$ and $Y$ be continuous variables then

Conditional Density Function of $Y$ given $X=x$ is defined by

$$f_{Y\mid X=x}(y)=\frac{f_{X,Y}(x,y)}{f_X(x)}$$

Conditional Distribution of the Bivariate Normal

Let $(X,Y)$ be bivariate normal with means ${\mu }_1,{\mu }_2$, variances ${\sigma }_1^2,{\sigma }_2^2$, and correlation $\rho$ then

$$Y\mid (X=x)\sim N\left({\mu }_2+\rho \frac{{\sigma }_2}{{\sigma }_1}(x-{\mu }_1),(1-{\rho }^2){\sigma }_2^2\right)$$