Probability generating functions
Generating functions
a sequence
,the generating function of the sequence, defined to be the sum of the power series # Integer-valued random variables Definition 4.7 The probability generating function (or pgf) of
is the function defined by for all values of s for which the right-hand side converges absolutely. and It is immediate that
exists for all values of s satisfying , since in this case Theorem 4.13 (Uniqueness theorem for probability generating functions) Suppose
and have probability generating functions and , respectively. Then if and only if ,for integer-valued random variables have the same probability generating function if and only if they have the same mass function
Here is a list of some common probability generating functions -
Bernoulli distribution:
Moments
- Definition 4.20 Let
. The moment of the random variable is the quantity . - Theorem 4.23 Let
be a random variable with probability generating function .The derivative of at equals for That is to say It is easy to see how to calculate the moments of from it. For example: and:
Sums of independent random variables
- Theorem 4.33 :If
and are independent random variables, each taking values in the set , then their sum has probability generating function - Theorem 4.36 (Random sum formula) Let
and be independent random variables, each taking values in . If the are identically distributed with common probability generating function , then the sum has probability generating function By this:
