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# Geometric Random Variables PowerPoint PPT Presentation

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.

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

• pmf: P(N=k) = e- k/k!

• Excel: POISSON(k,,TRUE/FALSE)

=3

=12.5

### 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…

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

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

### 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)

=1

=2

=4

### 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