Multivariate discrete distributions and independence

Bivariate discrete distributions

• Definition : If and are discrete random variables on, the joint (probability) mass function of and is the function defined by usually abbreviated to .
• It is clear that: and
• the marginal mass functions of and :

Expectation in the multivariate case

• If and are discrete random variables on and , it is easy to check that is a discrete random variable on also, defined formally by for . The expectation of may be calculated directly from the joint mass function , as the following theorem indicates:
• Theorem:We have that whenever this sum converges absolutely
• The expectation operator acts linearly on the set of discrete random variables:

Independence of discrete random variables

• Definition Two discrete random variables and are independent if the pair of events and are independent for all , and we normally write this condition as May be expressed as: Random variables which are not independent are called dependent
• This latter condition may be simplified as indicated by the following theorem:
• Theorem:Discrete random variables and are independent if and only if there exist functions such that the joint mass function of and satisfies:
• Theorem If and are independent discrete random variables with expectations and , then:
• Theorem:Discrete random variables and on (?, F , P) are independent if and only if: for all functions for which the last two expectations exist.
• Families of random variables with . For example, the family is called independent if: or, equivalently: Furthermore, if are independent, then Finally, the family is called pairwise independent if and are independent whenever

Sums of random variables

If and are discrete random variables with a certain joint mass function, what is the mass function of ? Clearly, takes the value if and only if and for some value of , and so If and are independent, their joint mass function factorizes, and we obtain the following result. - Theorem (Convolution formula):If and are independent discrete random variables on , then has mass function that the mass function of is the convolution of the mass functions of and .

Indicator functions

• Definition : The indicator function of an event is the random variable denoted and given by
• The function indicates whether or not occurs.It is a discrete random variable with expectation given by
• Indicator functions have two basic properties, namely: