P
Table of Contents
hw <= difficulty <= Jury Duty
1. Poisson
1.1. Poisson Process
- success (arrival) or failure
- # trials not fixed (continous)
- trials independent
- probabilities of success p very small. so we us λ = “average rate of arrivals in a given time frame”
1.2. More Perspective on the poisson process
The arrivals happen randomly and at an average rate of arrivals λ.
EX: 4 customers use a particular ATM per hour: 4 customers/hour
λ = 4 cusotmers/hour
1/λ = 1 customer/(1/4) hour on average…
1.3. How to know if its poisson?
Questions wiil usually be formated “Whats the probability that we’ll have x arrivals in the next(fixed) interval”
1.3.1. Probability Distribution
P(x= 3), P(2 < x < 6), P(λ > 2)
when x is poisson, we say x ~ poisson(λ)
p(X = x) = (λx * e(-λ))/x!, x = 0, 1, 2, …
let λ = 4 (customers/hour) we can plug in any x for number of events.
2. Practice Problems for Exam
2.1. 1
A communication system consists of 13 antennas arranged in a line. The system functions as long as no two nonfunctioning antennas are next to each other. Suppose five antennas stop functioning.
(a): How many different arrangements of the five nonfunctioning antennas result in the system being functional?
13 spaces and need to fill in 5
.0.0.0.0.0.0.0.0.
9 possible spaces = n
n = 9 r = 5
order doesnt matter (9*8*7*6*5)/5! = 126
(b): If the arrangement of the five nonfunctioning antennas is equally likely, what is the probability the system is functioning?
126/(13 choose 5)=.09