Math 3680

1
Math 3680 Lecture 5 Important
Discrete Distributions
2
The Binomial Distribution
3

• Example A student randomly guesses at three
  questions. Each question has five possible
  answers, only once of which is correct. Find the
  probability that she gets 0, 1, 2 or 3 correct.
  This is the same problem as the previous one we
  will now solve it by means of the binomial
  formula.

4

• Example Recall that if X Binomial(3, 0.2),
• P(X 0) 0.512
• P(X 1) 0.384
• P(X 2) 0.096
• P(X 3) 0.008
• Compute E(X) and SD(X).
•  

5

• MOMENTS OF Binomial(n, p) DISTRIBUTION
• E(X) n p
• SD(X)
• Var(X)
• Try this for the Binomial(3, 0.2) distribution.
• Do these formulas make intuitive sense?
•  

6

• Example A die is rolled 30 times. Let X
  denote the number of aces that appear.
• A) Find P(X 3).
• B) Find E(X) and SD(X).  

7

• Example Three draws are made with replacement
  from a box containing 6 tickets
• two labeled 1,
• one each labeled, 2, 3, 4 and 5.
• Find the probability of getting two 1s.

8
(No Transcript)
9
(No Transcript)
10
The Hypergeometric Distribution
11

• Example Three draws are made without
  replacement from a box containing 6 tickets
• two labeled 1,
• one each labeled, 2, 3, 4 and 5.
• Find the probability of getting two 1s.

12
P(S1S2S3) P(S1) P(S2 S1) P(S3 S1 n S2)
P(S1S2F3) P(S1) P(S2 S1) P(F3 S1 n S2)
P(S1F2S3) P(S1) P(F2 S1) P(S3 S1 n F2)
P(S1F2F3) P(S1) P(F2 S1) P(F3 S1 n F2)
P(F1S2S3) P(F1) P(S2 F1) P(S3 F1 n S2)
P(F1S2F3) P(F1) P(S2 F1) P(F3 S1 n F2)
P(F1F2S3) P(F1) P(F2 F1) P(S3 F1 n F2)
P(F1F2F3) P(F1) P(F2 F1) P(F3 F1 n F2)

13
(No Transcript)
14
The Hypergeometric Distribution

• Suppose that n draws are made without
  replacement from a finite population of size N
  which contains G good objects and B N - G
  bad objects. Let X denote the number of good
  objects drawn. Then
• where b n - g.

15

• Example Three draws are made without
  replacement from a box containing 6 tickets two
  of which are labeled 1, and one each labeled,
  2, 3, 4 and 5. Find the probability of
  getting two 1s.

16

• MOMENTS OF
• HYPERGEOMETRIC(N, G, n)
• DISTRIBUTION
• E(X) n p (where p G / N)
• SD(X)
• Var(X)

17

• REDUCTION FACTOR
• The term is called the Small
  Population
• Reduction Factor.
• It always appears when we draw without
  replacement.
• If the population is large (N gt 20 n) , then the
  reduction factor can generally be ignored (why?).

18

• Example
• Thirteen cards are dealt from a well-shuffled
  deck. Let X denote the number of hearts that
  appear.
• A) Find P(X 3).
• B) Find E(X) and SD(X).  

19
The Poisson Distribution
20
Example. A lonely bachelor decides to play the
field, deciding that a lifetime of watching
Leave It To Beaver reruns doesnt sound all
that pleasant. On 250 consecutive days, he calls
a different woman for a date. Unfortunately,
through the school of hard knocks, he knows that
the probability that a given woman will accept
his gracious invitation is only 1. What is
the chance that he will land three dates?
21
(No Transcript)
22
(No Transcript)
23
The Poisson Distribution

• If X is a discrete random variable that
  satisfies
• for m gt 0, then we say that X has a Poisson(m)
  distribution.
• Property This is a reasonable approximation of
  the Binomial(n, p) distribution if n gt 100 and n
  p lt 6. (In other words, n is large and p is very
  small.)

24

• Exercise Confirm that
• is a probability distribution for m gt 0.

25

• MOMENTS OF THE POISSON(m) DISTRIBUTION
• E(X) m
• SD(X)
• Var(X) m
• (In terms of the binomial distribution, why?)

26
Example It is estimated that one in two thousand
spectators at a sporting event will require first
aid treatment. Suppose that there are 11,000 fans
attending a particular event. Find the
probability that at least three spectators will
require treatment.
27
The Poisson Process
28

• A process is an experiment where events occur at
  random times, like the counts of a Geiger counter
  detecting radioactive decay. Although the decays
  occur at random times, the process seems to
  satisfy three conditions. Loosely stated, they
  are
• For a very short time interval, if we double the
  length of time, the probability of a decay during
  the time interval will double.
• In a short time interval, we are very unlikely
  to observe two or more decays.
• Just because we detected a decay during a
  one-second time interval, we do not expect it to
  be any more or less likely that we will detect
  another decay in the next one-second time
  interval.
• If a process satisfies these three conditions,
  then we call it a Poisson process.

29
Events like mutations in a population and
earthquakes can be modeled as a Poisson process.
However, a Poisson process may not be an
appropriate model for other events. For
instance, cars on a highway tend to cluster
behind a slowly moving car. If an event is the
passing of a car, then the third condition is not
satisfied. This is because if a car just passed,
then it is likely that more passes from the other
cars in the cluster will occur in the near
future.
30
The Poisson Process

• Suppose that a Poisson process has arrivals that
  occur at rate l per second (or other unit).
  Then the number of arrivals that occur in any
  interval with length t seconds has a Poisson(
  lt ) distribution.

31

• Example. In the Luria-Delbrück mutation model, it
  is assumed that the occurrences of mutations
  follow a Poisson process with an average of 0.25
  mutations per hour. Find the probability that at
  least one mutation occurs in the next two hours.