1. CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Discrete random variables Probability mass function Distribution function (Secs. 2.1-2.4)

2. Introduction  Sample space:  Associating real numbers with sample points:  Random variables:

3. Random variable: Example  Sequence of three coin tosses:  Sample space:  Random variable counts the number of heads in each sequence.

4. Random variable: Example  Random variable used to “reduce” the number of points in the sample space.

5. Discrete random variable  Definition:

6. Probability mass function  Mapping sample points to a random variable  Assigning probabilities to each one of the values X might take.  Counting the number of heads in a sequence of three coin tosses.

7. Probability mass function (contd..)  Properties of the probability mass function:

8. Distribution function  Difference between pmf and distribution function:  Example: Number of heads in three coin tosses:  P(a <= X <= b)

9. Distribution function (contd..)  Properties of distribution functions

10. Distribution function (contd..)  Distribution function of a discrete random variable:

11. Example: Majority voting system