1. CSE 531: Performance Analysis of Systems Lecture 2: Probs & Stats review Anshul Gandhi 1307, CS building anshul@cs.stonybrook.edu anshul.gandhi@stonybrook.edu 1

2. Outline 1.Announcements 2.Probability basics  Experiments, events, helpful relations 3.Random variables  Discrete  Bernoulli, Binomial, Geometric  Continuous  Uniform, Exponential 2

3. Announcements Collaborating on assignments Assignment 1 (next week) 3

4. Basics Probability is defined in terms of some experiment. The set of all outcomes of an experiment is its sample space. A subset of the sample space is called an event.  Mutually exclusive  Partition  Independent A function defined on the outcomes is a random variable. Law of total probability Conditional probability Bayes’ theorem 4

5. Random variables Discrete and Continuous Discrete  Countable possibilities  pmf 5

6. Discrete RVs PMF for sample space S  Pr[X = s] = p X (s) = p(s)   CDF: F X (a) = Pr[X ≤ a] =  Inverse CDF: F̅ X (a) = Pr[X > a] = 1 - F X (a) =  Mean E[X] =  E[X 2 ] =  Var[X] = E[X 2 ] – (E[X]) 2 6

7. Bernoulli(p) Outcome of a coin toss p(1) = p p(0) = 1-p  (find limits of s)  Mean E[X]  E[X 2 ]  Var[X] 7

8. Binomial(n, p) Number of 1’s when flipping a Bernoulli coin n times p(i) = n C i p i (1-p) (n-i)   Mean E[X]  E[X 2 ]  Var[X] 8

9. Geometric(p) Number of flips till we get a 1 p(i) = (1-p) (i-1). p   Mean E[X]  E[X 2 ]  Var[X] 9

10. Continuous RVs PDF for sample space S  Pr[a ≤ X ≤ b] =   CDF: F X (a) = Pr[X ≤ a] =   E[X i ] =  Var[X] = E[X 2 ] – (E[X]) 2 10

11. Uniform(a, b) f(x) = 1/(b-a) for a < x < b 11   E[X]  E[X 2 ]  Var[X]

12. Exponential(λ) f(x) = λ e - λ x, x ≥ 0 12   E[X]  E[X 2 ]  Var[X]