3. Joint and Conditional Distributions#

Errata#

As part of the continuous 2d example, the slide that appears at 10:18 in the lecture has an extra \(e\) in the denominator that shouldn’t be there. It appears to be a typo as it didn’t affect the final result. This is what that slide should look like, with the corrected term in red:

\[\begin{split} \begin{align*} f(x, y) &= \frac{1}{\pi} e^{-\frac{5}{2} \left( y^2 - \frac{2x}{5}y + \frac{x^2}{25} - \frac{x^2}{25} + \frac{x^2}{5} \right)}\\ &= \frac{1}{\pi} e^{-\frac{5}{2} \cdot \frac{4}{25} x^2} \cdot e^{-\frac{5}{2} \left(y - \frac{x}{5}\right)^2}\\ &= \frac{1}{\pi} \cdot e^{-\frac{2}{5}x^2} \cdot \sqrt{2\pi \cdot \frac{1}{5}} \cdot \frac{1}{\sqrt{2\pi \cdot \frac{1}{5}}} e^{-\frac{1}{2 \cdot \frac{1}{5}} \left(y - \frac{x}{5}\right)^2}\\ f(x) &= \frac{1}{\sqrt{2\pi \cdot \frac{5}{4}}} \textcolor{red}{e^{-\frac{x^2}{2 \cdot \frac{5}{4}}}} \cdot \int_{\mathbb{R}} \frac{1}{\sqrt{2\pi \cdot \frac{1}{5}}} e^{-\frac{1}{2 \cdot \frac{1}{5}} \left(y - \frac{x}{5}\right)^2} \, dy \end{align*} \end{split}\]

Joint Distributions#

A joint distribution describes the behavior of two or more random variables simultaneously. If \(X = (X_1, X_2, ..., X_n)\) is a vector of random variables, then the joint probability density function (PDF) \(f(x_1, x_2, ..., x_n)\) gives the likelihood that \(X_1\) takes the value \(x_1\), \(X_2\) takes the value \(x_2\), and so on.

The joint cumulative distribution function (CDF) \(F(x_1, x_2, ..., x_n)\) gives the probability that each of \(X_1, X_2, ..., X_n\) is less than or equal to \(x_1, x_2, ..., x_n\) respectively.

For two dimensions, i.e., when \(X = (X_1, X_2)\), we write these as:

  • Joint PDF: \(f(x_1, x_2)\)

  • Joint CDF: \(F(x_1, x_2)\)

Conditional Distributions#

A conditional distribution describes the behavior of one or more random variables given the values of some other variables. If we want to know the distribution of \(X_1\) given the value of \(X_2\), written \(f(x_1 | x_2)\), its defined as the ratio of the joint distribution to the marginal distribution of \(X_2\). The marginal distribution, \(f(x_2)\), is obtained by integrating the joint distribution over all values of \(X_1\):

\[f(x_1|x_2) = \frac{f(x_1, x_2)}{f(x_2)}, \quad \text{where} \quad f(x_2) = \int_{-\infty}^{\infty} f(x_1, x_2) dx_1\]