Download Discrete Probability Distributions

Discrete Probability Distributions
EGR 260
R. Van Til
Industrial & Systems Engineering Dept.
Copyright 2013. Robert P. Van Til. All rights reserved.
1
What’s It All About?
•  The behavior of many random processes can
be placed into a handful of categories.
–  In this presentation, we will develop probability
distributions for several common categories of
discrete random processes.
»  Note that not every discrete random process can be
modeled with one of these probability distributions.
•  In that case, you need to derive an appropriate probability
distribution (usually using counting principles).
–  In the next presentation, we will do the same for
continuous random processes.
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Definition
•  Bernoulli trials are a set of n trials of a random
process where the outcome of each trial is
–  Examples.
»  Roll a pair of dice
»  Flip a coin
–  A random process whose outcomes are Bernoulli
trials is said to “satisfy the Bernoulli property”.
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Binomial Distribution
•  Let RV X satisfy the Bernoulli property and be defined
by
X = {# of times of event A occurs in n trials}
then
where on any trial,
–  Note that order
4
Binomial Distribution
•  Where does this formula come from?
–  Run n trials of a Bernoulli process.
Determine the probability that event A occurs for the
first x trials and does not occur for the remaining n-x
trials. Call this event B1.
Since the trials are
»  Note any other arrangement of event A occurring x times
and Ac occurring n-x times has the same probability.
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Binomial Distribution
Suppose there are M different arrangements in which
event A occurs x times and Ac occurs n-x times (we
don’t yet know the value of M).
Let Bi, i=1,2,...,M, denote these M arrangements, then
P(x) = P(B1∪B2∪ ... ∪BM)
Since all Bi’s are
So, what’s the value of M?
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Binomial Distribution
There are n locations to place the x events A, the
remaining n-x locations will contain Ac.
So, M is the number of different ways to place x items
into n different locations and is given by
Hence,
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Aside
•  A combination, C(n,x), is the # of ways to select
x elements (without replacement) from a set of
n distinct elements where order does not
matter and is given by
–  Example: # ways to arrange 2 apples and 4 mangos
8
Example
•  Consider a injection molding machine which makes
interior trim components for cars. Define event A as
A = {machine makes a bad part}
Suppose that
P(A) = 0.05
and that the quality of each part is not effected by
those of the previous parts.
Determine the probability that 2 of the next 10 parts
produced are bad?
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Example
10
Properties of Binomial Distribution
•  The expected value and the variance for a
binomial distribution are given by
and
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CDF of a Binomial Distribution
•  The probability that event A occurs at most
j ≤ n times is
j
F(j) = ∑ C(n, x)p x q n-x
x=0
–  Recall F(j) is called the
€
–  Table II in the book’s appendices presents values of
F(j) for different values of j, n and p.
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Properties of Binomial Distribution
A typical binomial distribution (n = 5 & p = 0.35)
f(x)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
x
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Negative Binomial Distribution
•  Let the random process satisfy the Bernoulli property
and define RV X by
X = {trial # when event A occurs for the rth time}
then
where on any trial,
p = P(A) & q = P(Ac) = 1 - p
–  Note that order
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Negative Binomial Distribution
•  Where does this formula come from?
f(x) =
(x -1)!
=
p r q x-r
(r -1)!(x - r)!
€
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Special Case
•  For r=1, the negative binomial distribution is
termed the geometric distribution where
16
Example
•  The probability a CNC lathe makes a defective part is
0.02 for any part made by the lathe. Determine the
probability the lathe makes 22 parts before the 3rd
defective part is produced.
17
Properties of
Negative Binomial Distribution
•  The expected value and variance for a
negative binomial distribution are
E(X) = k/p
and
VAR(X) = kq/p2
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Hypergeometric Distribution
•  Consider a discrete sample space S where
N = # of elements in S & k = # of events A in S
Randomly sample n elements without replacement
from S and define the RV
x = {# times an event A is selected}
then
–  Note that
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Hypergeometric Distribution
•  Where does this formula come from?
1.  Number of ways to select n elements from a set
of N elements where order does not matter is
2.  Number of ways to select x events A from a set
containing k events A is
3.  Number of ways to select n-x events Ac from a
set containing N-k events Ac is
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Hypergeometric Distribution
•  From the counting formula,
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Example
•  Suppose a batch of 10 car engines contains 2 that are
defective. If 3 are selected at random without
replacement, what is the probability that 1 of the 3 is
defective?
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Example
•  What is the probability that all 3 engines
selected are defective?
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Properties of
Hypergeometric Distribution
•  The expected value and variance of a
hypergeometric distribution are
nk
E(X) =
N
and
nk " k %" N - n %
€VAR(X) = N $#1- N '&$# N -1 '&
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€
Binomial vs. Hypergeometric
Similarities
•  Sample n items, x of
which are event A and
remaining n-x are event
Ac.
•  Order
Differences
•  Binomial:
–  Bernoulli property
•  Hypergeometric:
–  Bernoulli property
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Binomial vs. Hypergeometric
•  If the size of the sample space N is large and
then probability f(x) is such that
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Example
•  Random process satisfies
Bernoulli property and
P(bad part)=0.05
What’s the probability
that 1 of 2 parts randomly
selected is bad?
•  Suppose 2 parts are
selected a random
without replacement from
a population of 100 parts
where 5 are bad.
What’s the probability
that 1 is bad?
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Another Aside
•  Suppose a set containing n elements has x1
elements of type 1, x2 elements of type 2, ...,
xM elements of type M. Then the # ways to
arrange all n elements of this set is
–  Note: n = x1 + x2 + ... + xM
»  Order matters among dissimilar types of elements, but it
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Example
•  How many different ways can you arrange 2
apples, 3 oranges and 4 mangos?
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Multinomial Distribution
•  Consider a discrete random process that satisfies the
Bernoulli property with k outcomes A1, A2, ..., Ak which
are all mutually exclusive and A1∪A2∪...∪Ak=S. Let
pi = P(Ai)
i = 1,2,...,k
and define the k RV’s
Xi = {# of times event Ai occurs in n trials} i=1,...,k
then
where x1+x2+...+xk=
and p1+p2+...+pk=
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Multinomial Distribution
•  The binomial distribution is a special case of
the multinomial distribution with
•  Note the multinomial distribution has k
random variables x1, x2, ..., xk.
–  Hence, our current definitions for expected value
and variance won’t work.
»  We will learn about these later when we study processes
with multiple RV’s.
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Example
•  Consider a CNC lathe which produces a part of
diameter d. Define events
A1 = {d ok}, A2 = {d too large}, A3 = {d too small}
where
P(A1) = 0.93,
P(A2) = 0.04,
P(A3) = 0.03
and that the quality of each part is not effected by
those of the previous parts.
Determine the probability that of the next 12 parts, 2
will be too large and 1 will be too small.
32
Example
33
General
Hypergeometric Distribution
•  Consider a sample space containing N discrete
elements. Each of these elements is classified as one
of J events A1, A2, ..., AJ where there are ki of each
event Ai and k1+k2+...+kJ=N.
Select n elements at random without replacement
and let the J RV’s be defined as
Xi = {# of times event Ai is selected}
i=1,...,J
then
–  Note that order
34
Example
•  Suppose there are 65 cars in a parking lot where 20
are Chevy’s, 15 are Fords, 17 are DaimlerChryslers
and 13 are Hondas.
If 10 cars are selected at random for emissions testing,
what is the probability that 4 are Chevy’s, 3 are
Fords, 2 are DaimlerChryslers and 1 is a Honda?
35
Example
36
Uniform Distribution
•  Consider a sample space with k distinct
elements denoted by the RV’s x1, x2, ..., xk. If
all k outcomes xi are equally likely to occur,
then
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Uniform Distribution
•  The expected value and variance for a
uniform distribution are given by
x1 + x 2 +... + x k
E(X i ) =
k
and
€
1 k
2
VAR(X i ) = ∑ (x i - E(X i ))
k i=1
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€
Poisson Process
•  A random process is called a Poisson process if an average
of λ events occur per unit time or unit space (e.g., unit
length, unit volume, etc.) and which satisfies:
1.  # of random events occurring in any segment of time or space is
independent of the number that occurred in previous segments
–  Called
2.  The average
3.  The smaller the segment of time or space, the lower the
probability of 2 random events occurring during that segment.
–  Hence, 2 or more random events cannot occur at the same time or
space.
39
Poisson Distribution
•  Let the discrete RV for a Poisson process be
defined as
X = {# of events that occur during a
specified time span t (or space )}
then
40
Example
•  A computer network receives an average of
0.1 messages/sec. and is a Poisson process.
Determine the probability that the number
of messages X during a 50 second interval is:
1.  Equal to 7.
2.  At least 4.
41
Example
42
Poisson Distribution
•  The expected value and variance for a
Poisson distribution are given by
E(X) = λt
and
VAR(X) = λt
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