Conditional probability

In probability theory, conditional probability describes the probability of event $A$ given that event $B$ has occurred. We define it as:

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$

Bayes’ theorem

An important result of conditional probability is Bayes’ theorem, which relates the conditional probability of an event $B$ given an event $A$ occurs. This allows us to compute the posterior probability. In the single event case:

$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$

In the multiple event case, where we have multiple disjoint sets $B_1,\cdots ,B_k$, then the probability of an event $B_i$ given event $A$ occurs is given by:

$$P(B_i|A)=\frac{P(B_i)P(A|B_i)}{{\sum }_{k=0}^{n}P(B_k)P(A|B_k)}$$

Random variables

Multiple random variables

In the continuous case, the conditional CDF is given by:

$$F_Y(y|x_k)=\frac{P(Y\le y_k,X=x_k)}{P(X=x_k)}\text{for}P(X=x_k)>0$$

And the conditional PDF is given by:

$$f_Y(y|x_k)=\frac{d}{dy}{F}_Y(y|x_k)=\frac{f_{XY}(x,y)}{f_X(x)}$$