CSE 531 Performance Analysis of Systems Lecture 2

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

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

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

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. Ø Mutually exclusive Ø Partition Ø Independent • A function defined on the outcomes is a random variable. • • • Law of total probability Conditional probability Bayes’ theorem 4

Random variables • Discrete and Continuous • Discrete Ø Countable possibilities Ø pmf 5

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

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

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

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

Continuous RVs • PDF for sample space S Ø Pr[a ≤ X ≤ b] = Ø Ø CDF: FX(a) = Pr[X ≤ a] = Ø Ø E[Xi] = Ø Var[X] = E[X 2] – (E[X])2 10

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

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