A simple [[Probability Density Function]] describes the probability law of a single continuous [[Random Variable]]. As an extension we can describe the combination of multiple continuous r.v’s. with a multi-dimensional joint PDF. This is analogous to the [[Joint Probability Mass Function]]. **Two-dimensional PDF:** $ f_{XY}(x,y)*\delta^2=\mathbf P(x \le X \le x+\delta, \, y \le Y \le y+\delta) $ Normalization to $1$ has to hold, when we integrate over all $x,y$ space. $ \int_{-\infty}^\infty \int_{-\infty}^\infty f_{X,Y}(x,y) \,dx\,dy=1 $ **Marginalization:** We can always reduce the multivariate PDF by a dimension. E.g. to get from $f_{X,Y}(x,y)$ to $f_X(x)$, we need to integrate over all possible $y$. $ f_X(x) =\int_{-\infty}^\infty f_{X,Y}(x,y)\,dy $ ![[pdf-marginalization.png|center|300]] We can prove the above by looking at the CDF $F_X(x)$. $ \begin{align} F_X(x) &= \int_{-\infty}^x \left(\int_{-\infty}^\infty f_{X,Y}(x,y) \,dy \right) \,dx \\[14pt] \underbrace{\frac{dF_X(x)}{dx}(x)}_{f_X(x)}&=\int_{-\infty}^\infty f_{X,Y}(x,y)\, dy \end{align} $ (1) We take the derivative on both sides. On the left side the derivative of a CDF is a PDF. On the right side, the derivative of an integral from $[-\infty, x]$ when integrating w.r.t. $x$ is the integrand itself. **Example:** We have a 2-dimensional space, while the $z$-axis represents the probability density. We want to calculate the volume of the rectangle $B$, which represents the probability where $a \le X \le b$ and $c \le Y \le d$. ![[joint-pdf.png|center|400]] $ \begin{align} \mathbf P \big((x,y) \in B\big) &=\int \int_{(x,y) \in B} f_{X,Y}(x,y)_dx_dy \\[12pt] &= \int_c^d \int_a^b f_{X,Y}(x,y) *dx *dy \end{align} $ When we calculate the volume of a very small rectangle with length and width $\delta$, it approximates to the following.. $ \underbrace{\mathbf P(x \le X \le x+\delta, y \le Y \le+y+\delta)}_{\text{probability}} \approx \underbrace{f_{X,Y}(x,y)}_{\text{density}}* \underbrace{\delta ^2}_{\text{unit area}} $ Therefore the density of a 2-dimensional PDF can be interpreted as probability per unit area. - $f_X(x)$ returns a probability per unit length on a one-dimensional line. - $f_{X,Y}(x,y)$ returns a probability per unit area on two-dimensional space. - etc.