Joint probability density

Consider two random variables $x,y$ . Their joint probability density $p(x_0,y_0) \dx \d y$ represents the probability that $x \in [x_0,x_0+\dx]$ and $y \in [y_0,y_0 + \d y]$ simultaneously.

We can define a cumulant similarly as for a single random variable,

P(x_0,y_0) = \int_{-\infty}^{x_0} \int_{-\infty}^{y_0} p(x,y)\d y \dx.

The joint probability can be related to the individual probabilities by

p(x,y)\dx \dy = p(y) \dy \times p(x|y) \dx.

The above relation can be used to derive Bayes’ theorem.