Probability Density Function - Densities Associated With Multiple Variables

Densities Associated With Multiple Variables

For continuous random variables X1, …, Xn, it is also possible to define a probability density function associated to the set as a whole, often called joint probability density function. This density function is defined as a function of the n variables, such that, for any domain D in the n-dimensional space of the values of the variables X1, …, Xn, the probability that a realisation of the set variables falls inside the domain D is

\Pr \left( X_1,\ldots,X_N \isin D \right) = \int_D f_{X_1,\dots,X_N}(x_1,\ldots,x_N)\,dx_1 \cdots dx_N.

If F(x1, …, xn) = Pr(X1 ≤ x1, …, Xn ≤ xn) is the cumulative distribution function of the vector (X1, …, Xn), then the joint probability density function can be computed as a partial derivative

f(x) = \frac{\partial^n F}{\partial x_1 \cdots \partial x_n} \bigg|_x

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