Joint Distributions

Joint Distributions#

A joint distribution for \(X\) and \(Y\) must determine all necessary information about distributions of \(X\) and \(Y\).

Cumulative Distribution Functions (CDFs)#

If \((X,Y)\) is a random vector and \(F\) is its cdf, then

  1. \(F\) is a non-decreasing function, meaning \(x \leq x'\) and \(y \leq y' \Rightarrow F(x,y) \leq F(x',y')\)

  2. \(F\) is ‘right-continuous’

  3. If \((x_n, y_n)\) is such that \( \text{min}(x_n, y_n) \rightarrow - \infty \), then \( F(x_n, y_n) \rightarrow 0 \)

  4. If \((x_n, y_n)\) is such that \( \text{min} (x_n, y_n) \rightarrow + \infty \), then \( F(x_n, y_n) \rightarrow 1 \)

  5. If \(x \leq x'\) and \(y \leq y'\), then \(F(x',y') - F(x',y) - F(x, y') + F(x, y) \geq 0\)

Marginal Distributions#

For discrete and continous cases, we can get marginal information (one-dimensional distribution for \(X\) or \(Y\)) from the joint pmf/pdf.

Discrete Cases#

\[ p_X(x) = \sum_y p_{X,Y}(x,y) \]
\[ p_Y(y) = \sum_x p_{X,Y}(x,y) \]

Continuous Cases#

\[ f_X(x) = \int_{-\infty}^\infty f_{X,Y}(x,y) \ dy \]
\[ f_Y(y) = \int_{-\infty}^\infty f_{X,Y}(x,y) \ dx \]

Expectations with Multiple Random Variables#

If \(V=h(X,Y)\), where \( h(X,Y): \mathbb{R}^2 \rightarrow \mathbb{R}^1 \), we don’t have to go through the pain of finding the distribution of \(V\) just to find \(E(V)\).

If \(X\) and \(Y\) are jointly discrete with pmf \(p_{X,Y}(x,y)\), provided \( \sum_{(x,y)} h(x,y) p_{X,Y}(x,y) \lt \infty \), we define:

\[ Eh(X,Y) = \sum_{(x,y)} h(x,y) p_{X,Y}(x,y) \]

Likewise, if \(X\) and \(Y\) are jointly continuous with pdf \( f_{X,Y}(x,y) \), provided \( \iint_{\mathbb{R}^2} h(x,y) f_{X,Y}(x,y) dydx \lt \infty \), where:

\[ Eh(X,Y) = \iint_{\mathbb{R}^2} h(x,y) f_{X,Y}(x,y) dydx \]

Expectations of functions with more than one random variable have the same linearity properties as functions of a single random variable:

  • \( E(ah(X,Y) + b) = a Eh(X,Y) + b \)

  • \( E(h(X,Y) + g(X,Y)) = Eh(X,Y) + Eg(X,Y) \)

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