Geometric Distribution Property • Geometric Distribution is memoryless – We want to repeat trials until 1st success occurs – Given that 1st success has NOT occurred, the conditional probability of the number of additional trials required (to get the 1st success) does NOT depend on the number of failures that have occurred until now.

IEOR @ IITBombay IE502: Probabilistic Models Geometric Distribution - pmf

p=0.7

p=0.5

p=0.3

IEOR @ IITBombay IE502: Probabilistic Models Discrete Uniform

• X ~ DiscreteUniform(x1, x2, .., xn) • X can take on any of the n possible outcomes, all of which are equally probable

• Example: Rolling a single die

IEOR @ IITBombay IE502: Probabilistic Models Poisson Distribution • Named after Siméon Denis Poisson • X ~ Poisson(λ)

• If expected number of occurrences in an interval is λ, then the probability that there are exactly k occurrences is given by Poisson distribution Distribution introduced in his book “Research on Probability of Judgments in Criminal and Civil Matters”

IEOR @ IITBombay IE502: Probabilistic Models Poisson Distribution pmf

λ=0.5

λ=2

λ=10

IEOR @ IITBombay IE502: Probabilistic Models Special Properties 1. Bernoulli distribution is a special case of Binomial distribution

IEOR @ IITBombay IE502: Probabilistic Models Special Properties

2. Suppose X1,…,Xn are independent and identically distributed (iid) random variables, all Bernoulli distributed

with parameter p, then the sum (X1+ X2 +…+ Xn) ~ Binomial(n, p)

IEOR @ IITBombay IE502: Probabilistic Models Special Properties 3. (Law of rare events) Suppose X ~ Binomial(n, p) where n is large and p is small, then distribution can be approximated as X ~ Poisson(np)

IEOR @ IITBombay IE502: Probabilistic Models Binomial Distribution - pmf

p=0.5, n =10

p=0.5, n =20

p=0.7, n =20

IEOR @ IITBombay IE502: Probabilistic Models Other Discrete Distributions • Negative Binomial Distribution • Hypergeometric Distribution

IEOR @ IITBombay IE502: Probabilistic Models Examples 1. Suppose an airplane engine will fail, when in flight, with probability 1-p independently from engine to engine; suppose that airplane will make a successful flight if at least 50% of its engines remain operative. For what values of p is a four- engine plane preferable to a two-engine plane?

2. An experiment consists of counting the number of persons arriving at an ATM in a 1 hour interval. Based on past data, we know that on an average 8.2 persons per hour arrive at the ATM. What is a good approximation to the probability that no more than 6 persons appear?

IEOR @ IITBombay IE502: Probabilistic Models Continuous Probability Distributions

IEOR @ IITBombay IE502: Probabilistic Models Uniform Distribution

• X ~ Uniform(0, 1) f(x) • X ~ U(a,b) – PDF

x • Mean and Variance a b

• Typically used when we have little knowledge of the process

IEOR @ IITBombay IE502: Probabilistic Models Exponential Distribution • X ~ Expo(λ) • PDF

• Mean & Variance

• Quite a popular distribution in modelling variety of situations: arrival processes, lifetimes, reliability models, queueing systems etc.

IEOR @ IITBombay IE502: Probabilistic Models Normal Distribution • Introduced by Abraham De Moivre • Later, Gauss & Laplace did extensive work on ‘Normal’ distribution • Many natural phenomena can be approximated by Normal distribution – Two parameter distribution – Bell shaped curve symmetric about the mean

• X ~ N(μ, σ2) – PDF

• Mean & Variance

• Standard Normal Distribution

IEOR @ IITBombay IE502: Probabilistic Models Other Continuous Distributions • Beta Distribution • Triangular Distribution • Gamma Distribution • Erlang Distribution • Weibull Distribution There are many • Log-Normal Distribution ‘special’ properties • Log-Logistic Distribution that relate different distributions • Chi-square Distribution • Student-t Distribution • F-Distribution • … etc etc

IEOR @ IITBombay IE502: Probabilistic Models