CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Multiple random variables Transform methods (Sec , 4.5.7)

Sum of random variables: Expectation  Expectation of a sum of two random variables:  Expectation of a sum of n random variables:  Special case: All the random variables have the same mean

Sum of random variables: Expectation  Alternate expression for variance:

Sum of random variables: Expectation  Example: Variance of the number of heads in a sequence of three coin tosses

Sum of random variables: Variance  Exponential random variable:

Sum of random variables: Variance

Moment generating function  Definition:

Moment generating function (contd..)  Laplace-Stilje’s transform:

Moment generating function (contd..)  Probability Generating Function:

Moment generating function (contd..)  How to use these functions:

Moment generating function (contd..)  Laplace-Stilje’s transform of the exponential distribution:

Moment generating function (contd..)  Convolution theorem:

Moment generating function (contd..)  Sum of exponential random variables (Erlang distribution):