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Geometric Random VariablesPowerPoint Presentation

Geometric Random Variables

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Geometric Random Variables

N ~ Geometric(p)

- # Bernoulli trials until the first success
- pmf: f(k) = (1-p)k-1p
- memoryless: P(N=n+k | N>n) = P(N=k)
- probability that we must wait k more coin flips for the first success is independent of n,the number of trials that have occurred so far

Previously…

- Conditional Probability
- Independence
- Probability Trees
- Discrete Random Variables
- Bernoulli
- Binomial
- Geometric

Agenda

- Poisson
- Continuous random variables:
- Uniform, Exponential

- E, Var
- Central Limit Theorem, Normal

Poisson

N ~ Poisson()

- N = # events in a certain time period
- average rate is
- Ex. cars arrivals at a stop sign
- average rate is 20/hr
- Poisson(5) = #arrivals in a 15 min period

Poisson

N1~Poisson(1), N2~Poisson(2)

- N1+N2 ~ Poisson(1+ 2)
- Splitting:
- Poisson() people arrive at L-stop
- probability p person is south bound
- Poisson(p) people arrive at L-stop south bound

other slides…

from Prof. Daskin’s slides

Xrandom variable

E[g(X)]=∑k g(k) P(X=k)

E[a X+b] = aE[X] +b

Var[a X + b] = a2 Var[X]

always

X1,…,Xn random variables

E[X1+…+ Xn] = E[X1]+…+E[Xn]

always

Var[X1+…+ Xn] = Var[X1]+…+Var[Xn]

when independent

E[X1·X2·…· Xn] = E[X1]·E[X2] ·…·E[Xn]

when independent

E and VarE[X]=p, Var[X]=p(1-p)

X~Binomial(N,p)

E[X]=Np, Var[X]=Np(1-p)

N~Geometric(p)

E[N]=1/p, Var[N]=(1-p)/p2

N~Poisson()

E[N]= , Var[N]=

X~U[a,b]

E[X]=(a+b)/2, Var[X]=(b-a)2/12

X~Exponential()

E[X]=1/, Var[X]=1/2

E, VarCentral Limit Theorem

X1,…,Xn i.i.d, µ=E[X1], 2=Var[X1]

- independent, identically distributed
Sn = X1,…,Xn

- E[Sn]=nµ, Var[Sn] = n2
- distribution approaches shape of Normal
- Normal(nµ,n2)

Normal Distribution

X1 ~ N(µ1,12), X2 ~ N(µ2,22)

- X1+X2 ~ N(µ1+µ2,12+22)
- pdf, cdf NORMALDIST(x,µ,,TRUE/FALSE)
- fractile / inverse cdf
- p=P(X≤z)
- NORMINV(p,µ,)

Newsvendor Problem

- must decide how many newspapers to buy before you know the day’s demand
- q = #of newspapers to buy
- b = contribution per newspaper sold
- c = loss per unsold newspaper
- random variable D demand

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