Chapter 9: Joint distributions and independence CIS 3033

9.1 Joint distributions: discrete Number of random variables: one, two, and more, especially when they are defined on the same sample space. [Otherwise consider the products of sample spaces, as in Section 2.4.]Section 2.4 What is new: influence between variables, as relation among events. For example, two random variables S and M, the sum and the maximum of two throws of a die.

9.1 Joint distributions: discrete The joint probability mass function of discrete random variables X and Y (on the same sample space Ω) is the function p: R 2 → [0, 1] defined by p(a, b) = P(X = a, Y = b) for −∞< a,b < ∞. The joint distribution function of random variables X and Y is the function F: R 2 → [0, 1] defined by F(a, b) = P(X ≤ a, Y ≤ b) for −∞< a,b < ∞. The marginal probability mass function of discrete random variables X or Y can be obtained from p(a, b) by summing the values of the other variable.

9.1 Joint distributions: discrete

In many cases the joint probability mass functions of X and Y cannot be retrieved from the marginal probability mass functions p X and p Y. This is also the case for the distribution functions.

9.2 Joint distributions: continuous

For the distribution functions, the relation is the same as the discrete case, as given in formula (9.1) and (9.2).

9.3 More than two random variables The joint distribution function F of X 1,X 2,..., X n (all defined in the same Ω) is defined by F(a 1, a 2,..., a n ) = P(X 1 ≤ a 1, X 2 ≤ a 2,..., X n ≤ a n ) for −∞ < a 1, a 2,..., a n < ∞. Joint probability mass function p can be defined for discrete random variables, and joint density function f can be defined for continues random variables, just like the case of two-variable.

9.3 More than two random variables Suppose a vase contains N balls numbered 1, 2,..., N, and we draw n balls without replacement. Since there are N(N−1) · · · (N−n+1) possible combinations for the values of X 1,X 2,..., X n, each having the same probability, the joint probability mass function is given by p(a 1, a 2,..., a n ) = P(X 1 =a 1, X 2 =a 2,..., X n =a n ) = 1 / [N(N − 1) · · · (N − n + 1)], for all distinct values a 1, a 2,..., a n with 1 ≤ a j ≤ N. The marginal distribution of each X i is p Xi (k) = 1/N.

9.4 Independent random variables Random variables X and Y are independent if every event involving only X is independent of every event involving only Y. Random variables that are not independent are called dependent. Random variables X and Y are independent if P(X ≤ a, Y ≤ b) = P(X ≤ a)P(Y ≤ b), that is, the joint distribution function F(a, b) = F X (a)F Y (b), for all possible values of a and b. The same conclusion applies to probability mass function p and density function f.

9.5 Propagation of independence