The moment generating function of random variable X is given by Moment generating function

The moment generating function of random variable X is given by Moment generating function

The moment generating function of random variable X is given by Moment generating function

More generally,

Example: X has the Poisson distribution with parameter

If X and Y are independent, then The moment generating function of the sum of two random variables is the product of the individual moment generating functions

Let Y = X 1 + X 2 where X 1 ~Poisson( 1 ) and X 2 ~Poisson( 2 ) and X 1 and X 1 are independent, then

Note: The moment generating function uniquely determines the distribution.

If X is a random variable that takes only nonnegative values, then for any a > 0, Markov’s inequality

Proof (in the case where X is continuous):

Let X 1, X 2,..., X n be a set of independent random variables having a common distribution, and let E[ X i ] = . then, with probability 1 Strong law of large numbers

Let X 1, X 2,..., X n be a set of independent random variables having a common distribution with mean  and variance  Then the distribution of Central Limit Theorem

Let X and Y be two discrete random variables, then the conditional probability mass function of X given that Y = y is defined as for all values of y for which P ( Y = y )>0. Conditional probability and conditional expectations

Let X and Y be two discrete random variables, then the conditional probability mass function of X given that Y = y is defined as for all values of y for which P ( Y = y )>0. The conditional expectation of X given that Y = y is defined as Conditional probability and conditional expectations

Let X and Y be two continuous random variables, then the conditional probability density function of X given that Y = y is defined as for all values of y for which f Y ( y )>0. The conditional expectation of X given that Y = y is defined as

Proof: