This document contains notes from a statistics lecture on discrete random variables. It discusses the concepts of the discrete uniform, binomial, and Poisson distributions. Examples are provided for each distribution. The lecturer takes questions from students regarding basketball scoring data modeled by Poisson distributions and how to calculate the probability of an event occurring. Simulations are demonstrated to check if a distribution fits real data. Students are reminded of properties to check for new distributions and key distributions to recognize. The next lecture will cover continuous random variables.
5. 1. Quiz!
2. Feedback
3. Discrete uniform distribution
4. More on the binomial distribution
5. Poisson distribution
Thursday, 4 February 2010
6. What you like
βββββββββββββββββββββ halfβcomplete proofs
ββββββββββββββββ video lecture
βββββββββββββ examples/applications
ββββββββ entertaining/engaging
ββββββββ your turn
βββββββ homework
ββββββ help sessions
βββββ website
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7. What youβd like
changed:
βββββββββ more examples/applications
βββββββ go slower
ββββββ calling on people randomly
ββββ less sex
βββ owlspace
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12. Discrete uniform
Equally likely events
f(x) = 1/(b - a) x = a, ..., b
X ~ DiscreteUniform(a, b)
What is the mean?
What is the variance?
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14. Your turn
In the 2008β09 season, Kobe Bryant attempted
564 ο¬eld goals and made 483 of them.
In the ο¬rst game of the next season he makes
7 attempts. How many do you expect he
makes? Whatβs the probability he doesnβt
make any? Whatβs the probability he makes
them all?
What assumptions did you make?
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21. Conditions
X = Number of times some event happens
(1) If number of events occurring in nonβ
overlapping times is independent, and
(2) probability of exactly one event
occurring in short interval of length h is β
Ξ»h, and
(3) probability of two or more events in a
sufο¬ciently short internal is basically 0
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22. Poisson
X ~ Poisson(Ξ»)
Sample space: positive integers
Ξ» β [0, β)
Arises as limit of binomial distribution when
np is ο¬xed and n β β = law of rare events.
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23. Examples
Number of alpha particles emitted from a
radioactive source
Number of calls to a switchboard
Number of eruptions of a volcano
Number of points scored in a game
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24. Basketball
Many people use the poisson distribution
to model scores in sports games. For
example, last season, on average the LA
lakers scored 109.6 points per game, and
had 101.8 points scored against them. So:
O ~ Poisson(109.6)
D ~ Poisson(101.8)
How can we check if this is reasonable?
http://www.databasebasketball.com/teams/teamyear.htm?tm=LAL&lg=n&yr=2008
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25. The data
0.08
0.06
Proportion
0.04
0.02
0.00
80 90 100 110 120 130
o
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27. Simulation
Comparing to underlying distribution
works well if we have a very large number
of trials. But only have 108 here.
Instead we can randomly draw 108
numbers from the speciο¬ed distribution
and see if they look like the real data.
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30. But
The distribution canβt be poisson. Why?
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31. Example
O ~ Poisson(109.6). D ~ Poisson(101.8)
On average, do you expect the Lakers to
win or lose? By how much?
Whatβs the probability that they score
exactly 100 points?
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32. Challenge
Whatβs the probability they score over 100
points? (How could you work this out?)
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34. Score difference
What about if weβre interested in the
winning/losing margin (offence β
defence)?
What is the distribution of O β D?
Itβs not trivial to determine. Weβll learn
more about it when we get to
transformations.
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35. Multiplication
X ~ Poisson(Ξ»)
Y = tX
Then Y ~ Poisson(Ξ»t)
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37. For a new distribution:
Compute expected value and variance
from deο¬nition (given partial proof to complete)
Compute the mgf (given random mathematical fact)
Compute the mean and variance from the
mgf (remembering variance isnβt second moment)
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39. Thursday
Continuous random variables.
New conditions & new deο¬nitions.
New distributions: uniform, exponential,
gamma and normal
Read: 3.2.3, 3.2.5
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