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-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

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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 mean 0

=1

=2

=4

Normal Distribution mean=0


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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