Todays Goals

1
Todays Goals

• Apply discrete probability distributions
• Poisson
• Homework 7 (due Wednesday March 25) Ch3 65 66
  (dont fall into the flaw of averages on (b)
  just find p( zero passengers) for (c) 68 77 1
  web problem.

2
Review

• Binomial
• What is the probability of x successes out of n
  trials.
• The probability p of a success is unaffected by
  previous successes.
• Hypergeometric
• What is the probability of x successes out of n
  trials
• sampling without replacement the probability of
  a success depends on what happened before
• Negative Binomial
• What is the probability of x failures before r
  successes?
• the probability p of a success is unaffected by
  previous successes.

3
Name the probability distribution and parameters

• A cable is composed of independent wires. When it
  is exposed to high overloads the probability that
  a wire will fracture is .05. The cable must be
  replaced when 3 wires have failed. What is the
  probability that the cable can withstand at least
  5 overloads before being replaced?
• Binomial
• Hypergeometric
• Geometric
• Negative Binomial

4
Name the probability distribution and parameters

• A cable is composed of independent wires. When it
  is exposed to high overloads the probability that
  a wire will fracture is .05. The cable must be
  replaced when 3 wires have failed. What is the
  probability that the cable can withstand at least
  5 overloads before being replaced?
• Negative binomial
• r 3
• p .05
• Y total number of overloads
• X number of overloads where no wires break
• Y X 3
• P(Ygt5) P(X gt 2) P(X3)

5
Negative Binomial Distribution

• The probability that is takes Xx failures to get
  r successes, with probability of success p is

6
Poisson Process

• Often we are interested in events that can occur
  at any point in time or space
• Fatigue cracks may occur continuously along a
  weld
• earthquakes may occur anytime, and anywhere over
  a seismically active region
• Traffic accidents can happen anywhere on a given
  highway
• We can model such occurrences with a Poisson
  Process

7
Poisson Process

• An event can occur at random and at any time (or
  point in space)
• The occurrence(s) of an event in a fixed time (or
  space) interval is independent of that in any
  other non-overlapping interval
• The probability of occurrence of an event in a
  small interval Dt, is proportional to Dt and can
  be given by aDt, a is the mean rate of occurrence
  of the event.

8
Poisson Random Variable

• Let Xt be the number of occurrences in the time
  (or space) interval t where the mean rate of
  occurrence is a. Then
• If X has a Poisson distribution with parameter
  lat, then

9
Example

• Suppose that in a previous traffic count we
  observed an average of 60 cars per hour making
  left turns. What is the probability that exactly
  5 cars make a left turn in a 10 minute interval?

10
Example

• Suppose that in a previous traffic count we
  observed and average of 60 cars per hour making
  left turns. What is the probability that exactly
  10 cars make a left turn in a 10 minute interval?
• Poisson with a 60 (per hour) and t 1/6 hour.
• Or, a 1 (per minute) and t 10 minutes

11
Example

• Suppose that in a previous traffic count we
  observed and average of 60 cars per hour making
  left turns. What is the probability that exactly
  10 cars make a left turn in a 10 minute interval?
• Poisson with a 60 (per hour) and t 1/6 hour.
• Or, a 1 (per minute) and t 10 minutes

12
Example

• Suppose that the average number of accidents
  occurring weekly on a particular stretch of the
  highway equals 3. Calculate the probability that
  there is at least one accident this week.

13
Example

• Suppose that the average number of accidents
  occurring weekly on a particular stretch of the
  highway equals 3. Calculate the probability that
  there is at least one accident this week.

l 3
14
Poisson as limiting distribution of Binomial

• Suppose that in the binomial pmf b(xn,p), we let
  n ? 8 and p ? 0 in such a way that np approaches
  a value l gt 0. Then
• As a rule of thumb, n ? 100, p lt .01, and
  np lt 20.

15
Poisson as limiting distribution of Binomial --
example

• About 1 in 10,000 people has a rare disease.
• In a building of 500 people, what is the expected
  number with the disease?

16
Poisson as limiting distribution of Binomial --
example

• About 1 in 10,000 people has a rare disease. What
  is the probability that 5 people in a building of
  500 people would have this disease? Probability
  that one or more would have it?
• Use Poisson approximation.

17
Poisson as limiting distribution of Binomial --
example

• About 1 in 10,000 people has a rare disease. What
  is the probability that 5 people in a building of
  500 people would have this disease? Probability
  that one or more would have it?
• Use Poisson approximation.
• lambda np .0001500 .05
• p(5)
• p(1 or more) 1-p(0) 1-

18
Poisson as limiting distribution of Binomial --
example

• About 1 in 10,000 people has a rare disease. What
  is the probability that 5 people in a building of
  500 people would have this disease? Probability
  that one or more would have it?
• Compare with exact
• p(5) 2.4310-9
  (2.510-9)

19
Example

• Automobiles arrive at a vehicle equipment
  inspection station according to a Poisson process
  with rate 10 per hour.
• What is the probability that exactly 1 vehicle
  arrives during the first 6 minutes?

20
Example

• Automobiles arrive at a vehicle equipment
  inspection station according to a Poisson process
  with rate 10 per hour.
• What is the probability that exactly 1 vehicle
  arrives during the first 6 minutes?
• a 10/hour t 1/10 hour l 101/10 1

21
Example

• Automobiles arrive at a vehicle equipment
  inspection station according to a Poisson process
  with rate 10 per hour.
• Suppose that with probability .5 an arriving
  vehicle will have no equipment violations.
• What is the probability that exactly 1 vehicle
  arrives during the first 6 minutes and has no
  violations?

22
Example

• Automobiles arrive at a vehicle equipment
  inspection station according to a Poisson process
  with rate 10 per hour.
• Suppose that with probability .5 an arriving
  vehicle will have no equipment violations.
• What is the probability that exactly 1 vehicle
  arrives during the first 6 minutes and has no
  violations? .37.5 18.5

23
Example Design of left turn bay

• The cycle time of the traffic light for left
  turns is 1 minute. The design criterion requires
  a left turn lane that will be sufficient 96 of
  the time. What should the length of the left turn
  bay be (in terms of car lengths) to allow for an
  average of 100 left turns per hour?

24
Example Design of left turn bay

• The cycle time of the traffic light for left
  turns is 1 minute. The design criterion requires
  a left turn lane that will be sufficient 96 of
  the time. What should the length of the left turn
  bay be (in terms of car lengths) to allow for an
  average of 100 left turns per hour?
• Let X number of cars that arrive to make a left
  turn during the 1 minute cycle time.
• Let the length of the turn bay be k car lengths
• Want P(Xk) 0.96

25
Example Design of left turn bay

• The cycle time of the traffic light for left
  turns is 1 minute. The design criterion requires
  a left turn lane that will be sufficient 96 of
  the time. What should the length of the left turn
  bay be (in terms of car lengths) to allow for an
  average of 100 left turns per hour?
• Let X number of cars that arrive to make a left
  turn during the 1 minute cycle time.
• Let the length of the turn bay be k car lengths
• Want P(Xk) 0.96
• Cars arrive at a rate of 100/60 1.67 per
  minute.

26
So, the left turn bay should be 4 car lengths
long to meet the criterion.
27
Discrete Uniform Random Variable

• A Discrete Uniform r.v. on 1,2,,n assigns
  equal probability to each of those integer
  values.
• p(x) 1/n for x1,2,,n.