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Geometric Random Variables. N ~ Geometric(p) # Bernoulli trials until the first success pmf: f(k) = (1-p) k-1 p 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.

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geometric random variables
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
Previously…
  • Conditional Probability
  • Independence
  • Probability Trees
  • Discrete Random Variables
    • Bernoulli
    • Binomial
    • Geometric
agenda
Agenda
  • Poisson
  • Continuous random variables:
    • Uniform, Exponential
  • E, Var
  • Central Limit Theorem, Normal
poisson
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
poisson1
Poisson
  • pmf: P(N=k) = e- k/k!
  • Excel: POISSON(k,,TRUE/FALSE)

=3

=12.5

poisson2
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
other slides…

from Prof. Daskin’s slides

e and var
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 Var
e var
X~Bernoulli(p)

E[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, Var
central limit theorem
Central Limit Theorem

X1,…,Xn i.i.d, µ=E[X1], 2=Var[X1]

  • independent, identically distributed

Sn = X1,…,Xn

  • E[Sn]=nµ, Var[Sn] = n2
  • distribution approaches shape of Normal
    • Normal(nµ,n2)
normal distribution
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
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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