Joint Probability

Joint Probability#

The joint probability distribution describes the chance of the values of multiple random variables. The joint distribution is denoted as,

\[ P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n) \]

  • Equivalently the comma can be replaced with interesection \(\cap\) or “logical and” \(\land\)

There are many different ways to write joint events:

  • Random variables take singular values,

    \[ P(X=x, Y=y) \]

  • Equality and inequality of random variables,

    \[ P(g(X) \ge h(Y)) \]

  • In general all joint distribution can be described by set of all possible values that makes the event true,

    \[ \begin{gather} P((X,Y) \in S), S = \set{(x_1,y_1), (x_2,y_2), \ldots} \end{gather} \]

Derived from Conditional Probability#

Take two random variables \(X\) and \(Y\). The joint probability for some value \(x,y\) can be determined by the relationship with the conditional probability,

\[\begin{split} \boxed{\begin{align*} P(X=x,Y=y) &= P(X=x \mid Y=y)P(Y=y)\\ &= P(Y=y \mid X=x)P(X=x) \end{align*}} \end{split}\]

The equality of the two lines comes from the divisoin rule or Bayes rule:

\[\begin{split} P(X=x \mid Y=y) = \frac{P(X=x, Y=y)}{P(Y=y)}\\ P(Y=y \mid X=x) = \frac{P(X=x, Y=y)}{P(Y=y)} \end{split}\]

In general the probability of a joint probability is given by the chain rule,

\[ \boxed{P\left(\bigcap_{i=1}^n X_i\right) = \prod_{i=1}^n P\left(X_i \Bigg\lvert \bigcap_{j\neq i}^{n}X_j \right)}\]

Independent Joint Probability#

Random variables are considered independent if the their probability distribution are separable. For instance for \(X\) and \(Y\),

\[ P_{XY}(x,y) = P_X(x)P_Y(y) \]

A more compact notation,

\[ P(x,y) = P(x)P(y) \]

Alternatively two random variables are independent if the conditional probability is the probability of itself,

\[ P(x \mid y) = P(x) \]

Marginal Probability#

The marginal probability is a posterior probability summing over all joint probability for each prior event. For instance given that \(X\) is a random variable of the posterior and \(Y\) of the prior

\[ P_X(x) = \sum_{y \in \Omega_Y} P(x, y) = \sum_{y \in \Omega_Y}{P(x \mid y)P(y)} \]

Expectation Value#

The expectation value of a joint probability follows linearity such that,

\[ \mathbb E[aX+bY+c] = \sum_{x \in X}{\sum_{y \in Y}}{P(x,y)(ax+by+c)} \]

Union Probability#

Inclusion-Exclusion Formula#

The inclusion exclusion formula defines the probability of the union of two or more events.

Let’s start simply with the union of two events which states:

**"The probability of any event of $A$ or $B$ occuring."**
\[ P(A \cup B) = P(A) + P(B) - P(A, B) \]

In general,

\[ P\left(\bigcup_{i=1}^n A_i\right) = \sum_i^n P(A_i) - \mathop{\sum\sum}\limits_{1\le i < j \le n}P(A_iA_j) + \ldots + (-1)^{n+1}P(A_1,A_2, \ldots, A_n) \]

Boole’s Inequality#

Boole’s Inequality is an upper bound for the union of two or more events. It’s simply says that the probability of a collection of event is no greater than the sum of all its probabilities,

\[ P\left(\bigcup_{i=1}^nA_i\right) \le \sum_{i=1}^{n} P(A_i) \]
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