Download 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…
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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)
0.12
1
1
0.9
0.9
0.8
0.15
0.6
0.5
0.1
0.4
0.3
0.05
0.2
Cum Prob
Probability
0.7
Probability
0.2
0.1
0.8
0.7
0.08
0.6
0.06
0.5
0.4
0.04
0.3
0.2
0.02
0.1
0.1
0
0
0
5
10
15
Number
=3
20
25
0
0
0
5
10
15
Number
=12.5
20
25
Cum Prob
0.25
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
E and Var
X random 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, 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)
Normal Distribution mean=0
Normal Distn
=1
=2
=4
-15
-10
-5
0
5
10
15
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