Chapter 4Chapter 4

Discrete Probability Discrete Probability DistributionsDistributions

Roll Two Dice and Record the SumsRoll Two Dice and Record the Sums

Physical Outcome:Physical Outcome:

An ordered pair ofAn ordered pair of

two faces showing.two faces showing.

We assign aWe assign a

numeric value tonumeric value to

each pair byeach pair bycounting up all ofcounting up all of

the dots that show.the dots that show.

A Function of EventsA Function of Events Note that there may beNote that there may be

several outcomes thatseveral outcomes thatget the same value.get the same value.

This assignment of aThis assignment of anumeric value is, in fact,numeric value is, in fact,a function.a function.

The domain is a set The domain is a set containing the possiblecontaining the possibleoutcomes, and the rangeoutcomes, and the rangeis the set of numbersis the set of numbersthat are assigned to thethat are assigned to theoutcomes.outcomes.

You might say, for example,You might say, for example,f( )=5.f( )=5.

Random VariableRandom Variable However, we don’t use f(x) notation in this case.However, we don’t use f(x) notation in this case. This function is called a This function is called a random variablerandom variable and is and is

typically given a capital letter name, such as X.typically given a capital letter name, such as X. Even though it is truly a function, we use it much Even though it is truly a function, we use it much

the same way that we would a variable in algebra, the same way that we would a variable in algebra, except for one thing:except for one thing:

The random variable takes on different values The random variable takes on different values according to a probability distribution associated according to a probability distribution associated with the underlying events.with the underlying events.

That is, we can never be certain what the value That is, we can never be certain what the value will be (random) andwill be (random) and

The values vary (variable) from trial to trial.The values vary (variable) from trial to trial.

The Probabilities of XThe Probabilities of X We have already used the notation P(A) in We have already used the notation P(A) in

connection with the probabilities of events.connection with the probabilities of events. P is a function that relates an P is a function that relates an eventevent to a to a probabilityprobability

(a number between 0 and 1).(a number between 0 and 1). Similarly, we will use expressions like Similarly, we will use expressions like P(X=x)=.5P(X=x)=.5, or , or

P(X=3)=.5.P(X=3)=.5. Why not just say P(3)=.5? We sometimes do, for Why not just say P(3)=.5? We sometimes do, for

short. Technically, 3 doesn’t have a probability. short. Technically, 3 doesn’t have a probability. It’s the event, X=3, that has a probability.It’s the event, X=3, that has a probability.

X=3 should be understood as X=3 should be understood as “X takes the value 3”,“X takes the value 3”, rather than “X equals 3”. rather than “X equals 3”.

P(X=3)=.5 says 3 is a value of X that occurs with P(X=3)=.5 says 3 is a value of X that occurs with probability .5.probability .5.

Lower-case letters are used for particular values, Lower-case letters are used for particular values, upper-case for r.v. names, as in P(X=x).upper-case for r.v. names, as in P(X=x).

A Random A Random Variable X Variable X for two dicefor two dice

The table lists the The table lists the outcomes that are outcomes that are mapped to each mapped to each sum, x.sum, x.

The n(x) column tells The n(x) column tells how many equally how many equally likely outcomes are likely outcomes are in each group.in each group.

P(X=x) = n(x)/n(S) P(X=x) = n(x)/n(S) = n(x)/36. = n(x)/36.

OutcomesOutcomes xx n(x)n(x)

(1,1)(1,1) 22 11

(2,1),(1,2)(2,1),(1,2) 33 22

(3,1),(2,2),(1,3)(3,1),(2,2),(1,3) 44 33

(4,1),(3,2),(2,3),(1,4)(4,1),(3,2),(2,3),(1,4) 55 44

(5,1),(4,2),(3,3),(2.4),(1,5)(5,1),(4,2),(3,3),(2.4),(1,5) 66 55

(6,1),(5,2),(4,3),(3,4),(2,5),(1,6)(6,1),(5,2),(4,3),(3,4),(2,5),(1,6) 77 66

(6,2),(5,3),(4,4),(3,5),(2,6)(6,2),(5,3),(4,4),(3,5),(2,6) 88 55

(6,3),(5,4),(4,5),(3,6)(6,3),(5,4),(4,5),(3,6) 99 44

(6,4),(5,5),(4,6)(6,4),(5,5),(4,6) 1010 33

(6,5),(5,6)(6,5),(5,6) 1111 22

(6,6)(6,6) 1212 11

Probability Histogram for XProbability Histogram for X

A Probability Function Definition A Probability Function Definition A histogram is often called a “distribution” because it A histogram is often called a “distribution” because it

graphically depicts how the probability is distributed among graphically depicts how the probability is distributed among the values. (Actually, a histogram is just a picture of a the values. (Actually, a histogram is just a picture of a distribution, not the distribution itself.) distribution, not the distribution itself.)

We also like to have a formula that gives us the probability We also like to have a formula that gives us the probability values when this is possible.values when this is possible.

The 2-dice toss problem gives a nice regular shape. Can The 2-dice toss problem gives a nice regular shape. Can we come up with a formula for the probabilities?we come up with a formula for the probabilities?

It is a V-shaped function, which is typical of absolute value It is a V-shaped function, which is typical of absolute value graphs. Since the vertex is at x=7, we could try something graphs. Since the vertex is at x=7, we could try something with |x-7|. A little experimentation will lead towith |x-7|. A little experimentation will lead to

6 | 7 | if {2,3,4,...,12}

( ) 360 otherwise

xx

P X x

Even EasierEven Easier Consider the toss of a single die.Consider the toss of a single die.

– Define a random variable X as the number of spots that Define a random variable X as the number of spots that show on the top face.show on the top face.

– Define a probability function for X as Define a probability function for X as

Consider a coin toss. LetConsider a coin toss. Let

Define a probability function for X as Define a probability function for X as

1 if {1,2,3,4,5,6}

P( ) 60 otherwise

xX x

1 if {0,1}

P( ) 20 otherwise

xX x

0 if the outcome is tails

1 if the outcome is heads X

Measures of Central TendencyMeasures of Central Tendency Find the mean of a distributionFind the mean of a distribution Think: What is a mean?Think: What is a mean?

– Average of all observationsAverage of all observations– Theoretical long run average of Theoretical long run average of

observationsobservations Calculate this from the information in Calculate this from the information in

the probability distributionthe probability distribution

Example of a Simple Probability DistributionExample of a Simple Probability Distribution

Say we have a discrete r.v. X as follows:Say we have a discrete r.v. X as follows:

Suppose we have 10 realizations of X. If Suppose we have 10 realizations of X. If the 10 occurred in the exact long-run the 10 occurred in the exact long-run proportions, what would they be?proportions, what would they be?1, 1, 1, 2, 2, 2, 2, 3, 3, 3.1, 1, 1, 2, 2, 2, 2, 3, 3, 3.

xx P(X=x)P(X=x)

11 .3.3

22 .4.4

33 .3.3

Calculate the MeanCalculate the Mean What then would the mean be?What then would the mean be?

Note: The mean doesn’t have to be a value of Note: The mean doesn’t have to be a value of X.X.

All

(1 1 1 2 2 2 2 3 3 3) /10

1 3 2 4 3 3

10 10 101 .3 2 .4 3 .3 2

1 P(1) 2 P(2) 3 P(3)

P( )x

x x

Expected Value of a Discrete R.V.Expected Value of a Discrete R.V.

All

E( ) P( )x

X x x

All

E( ( )) ( ) P( )x

f X f x x

Variance of a Discrete R.V.Variance of a Discrete R.V.

Variance is also an expected valueVariance is also an expected value

Standard Deviation, as always, is the Standard Deviation, as always, is the square root of the variance.square root of the variance.

2 2 2

All

2 2

var( ) E[( ) ] ( ) P( )

[ P( )]

x

X X x x

x x

2

Example: The number of standby passengers who get seats on a daily commuter flight from Boston to New York is a random variable, X, with probability distribution given below. Find the mean, variance, and standard deviation.

0 0.30 0.00 -1.55 0.72081 0.25 0.25 -0.55 0.07562 0.20 0.40 0.45 0.04053 0.15 0.45 1.45 0.31544 0.05 0.20 2.45 0.30015 0.05 0.25 3.45 0.5951

Totals 1.00 1.55 2.0475

x P x( ) xP x( ) x )()( 2 xPx

Solution:

Using the formulas for mean, variance, and standard deviation:

Note: 1.55 is not a value of the random variable (in this case). It is only what happens on average.

[ ( )] .xP x 155

2 2 0475 143. .

0475.2)()( 22 xPx

Example: The probability distribution for a random variable x is given by the probability function

Find the mean, variance, and standard deviation.

Solution:

Find the probability associated with each value by using the probability function.

P xx

x( ) 815

for 3, 4, 5, 6, 7

P( )38 315

515

P( )48 415

415

P( )58 515

315

P( )68 615

215

P( )78 715

115

3 5/15 15/15 -4/3 80/1354 4/15 16/15 -1/3 4/1355 3/15 15/15 2/3 12/1356 2/15 12/15 5/3 50/1357 1/15 7/15 8/3 64/135

Totals 15/15 65/15 210/135

x P x( ) xP x( ) x )()( 2 xPx

33.43

13

15

65)]([ xxP

2 156 125. .

56.1135

210)P()( 22 xx

Binary ExperimentsBinary Experiments(Bernoulli Trials)(Bernoulli Trials)

A Bernoulli Trial is an experiment for which A Bernoulli Trial is an experiment for which there are only two possible outcomes.there are only two possible outcomes.

For probability theory purposes, these are For probability theory purposes, these are designated “success” and “failure,” designated “success” and “failure,” although the names are arbitrary.although the names are arbitrary.

Examples include a coin toss with Examples include a coin toss with outcomes of heads or tails, or any outcomes of heads or tails, or any experiment where the results are yes or experiment where the results are yes or no, true or false, good or defective, etc.no, true or false, good or defective, etc.

The Bernoulli DistributionThe Bernoulli Distribution

A Bernoulli R.V. assigns to each outcome of a A Bernoulli R.V. assigns to each outcome of a Bernoulli Trial a 1 for success or a 0 for failure.Bernoulli Trial a 1 for success or a 0 for failure.

P(1)P(1) is denoted by is denoted by pp and is the parameter of the and is the parameter of the distribution (probability of a success).distribution (probability of a success).

P(0)=(1–p)P(0)=(1–p) because {0} is the complement of {1}. because {0} is the complement of {1}. The notation The notation q=(1-p)q=(1-p) is also used to simplify is also used to simplify

formulas. However, q is not another parameter, formulas. However, q is not another parameter, because its value is determined by p.because its value is determined by p.

Mean and Variance of BernoulliMean and Variance of Bernoulli

E( )

(0) P(0) (1) P(1)

0

X

p

p

2 2

2 2

2 2

E( )

(0 ) P(0) (1 ) P(1)

(1 ) (1 )

(1 )( 1 )

(1 )

X p

p p

p p p p

p p p p

p p

pq

Some ExamplesSome Examples

Bernoulli Trials p=0.5

0

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Bernoulli Trials p=0.25

0

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Binomial DistributionBinomial Distribution

Suppose in a Suppose in a series ofseries of nn Bernoulli Trials you Bernoulli Trials you keep track of the total number of successes.keep track of the total number of successes.

The trials are independent.The trials are independent. We say We say pp and and nn are the parameters of the are the parameters of the

distribution.distribution. Let X be a r.v. for the number of successes.Let X be a r.v. for the number of successes. Let’s start with n=2 and p is 1/4. Let’s start with n=2 and p is 1/4. The next slide shows the outcomes with The next slide shows the outcomes with

corresponding values of X and probabilities.corresponding values of X and probabilities.

Binomial Probability ExampleBinomial Probability Example

OutcomeOutcome ProbabilityProbability Value of XValue of X (SS) (SS) (1/4)(1/4)=1/16 (1/4)(1/4)=1/16 22 (SF) (SF) (1/4)(3/4)=3/16 (1/4)(3/4)=3/16 11 (FS) (FS) (3/4)(1/4)=3/16 (3/4)(1/4)=3/16 11 (FF) (FF) (3/4)(3/4)=9/16 (3/4)(3/4)=9/16 0 0

Value of XValue of X Probability ofProbability of22 1/16 1/1611 6/16 6/1600 9/16 9/16

Mean and Variance of Binomial Mean and Variance of Binomial when n=2when n=2

E( )

0 P(0) 1 P(1) 2 P(2)

9 6 10 1 2

16 16 168 1

16 2

X

np

2 2

2 2 2

E( )

1 1 10 P(0) 1 P(1) 2 P(2)

2 2 2

1 9 1 6 9 1

4 16 4 16 4 1624 3

64 8

X

npq

Binomial ProbabilitiesBinomial Probabilities

For a binomial r.v. X, with probability For a binomial r.v. X, with probability parameter p and n trials,parameter p and n trials,

Example: Calculate the probability of Example: Calculate the probability of 3 successes in 5 trials if p=.25.3 successes in 5 trials if p=.25.

P( ) (1 )x n xnx p p

x

3 2

5 4

5 1 3P(3)

3 4 4

5! 9 5 4 3 2 9 45

3!2! 4 3 2 2 4 4 128

A Fishing TripA Fishing Trip Dr. A is fond of fly-fishing in the Colorado mountain Dr. A is fond of fly-fishing in the Colorado mountain

streams. His long-run average is one catch per 20 casts. streams. His long-run average is one catch per 20 casts. On the last day of his vacation, He stops to eat supper. On the last day of his vacation, He stops to eat supper. He figures he will cast 10 more times and then pack up. He figures he will cast 10 more times and then pack up. What is the probability he will catch one more fish?What is the probability he will catch one more fish?

What is the probability he will catch What is the probability he will catch at leastat least one more fish? one more fish?

What is the probability he will catch What is the probability he will catch more thanmore than one fish? one fish?

0 10 1010P( 1) 1 P( 0) 1 .05 .95 1 .95 .401

0X X

1 9 910P( 1) .05 .95 10 .05 .95 .315

1X

P( 1) P( 1) P( 1) .401 .315 .086X X X

From Binomial to PoissonFrom Binomial to Poisson The The Binomial DistributionBinomial Distribution deals with counting deals with counting

“successes” in a “successes” in a fixed numberfixed number of trials. It is of trials. It is discrete and finite-valueddiscrete and finite-valued..

Suppose there is Suppose there is no specified number of trialsno specified number of trials. . We may want to count the We may want to count the number of “successes”number of “successes” that occur that occur in an intervalin an interval of time or space. of time or space.

““Successes,” or sometimes “events,” (not to be Successes,” or sometimes “events,” (not to be confused with probability events) refer to confused with probability events) refer to whatever we are interested in counting, such aswhatever we are interested in counting, such as– the number of errors on a typed page,the number of errors on a typed page,– the number of flights that leave an airport in an hour,the number of flights that leave an airport in an hour,– the number of people who get on the bus at each stop,the number of people who get on the bus at each stop,– or the number of flaws on the surface of a metal sheet.or the number of flaws on the surface of a metal sheet.

Poisson DistributionPoisson Distribution A Poisson Random Variable, X, takes on A Poisson Random Variable, X, takes on

values 0, 1, 2, 3, . . . , corresponding to the values 0, 1, 2, 3, . . . , corresponding to the number of events that occur.number of events that occur.

Since no definite upper bound can be Since no definite upper bound can be given, X is an given, X is an infinitely-valued discreteinfinitely-valued discrete random variable.random variable.

Assumptions:Assumptions:– The probability that an event occurs is the The probability that an event occurs is the

same for each unit of time or space.same for each unit of time or space.– The number of events that occur in one unit of The number of events that occur in one unit of

time or space is independent of any others.time or space is independent of any others.

Poisson Probability FunctionPoisson Probability Function

The probability of observing exactly x The probability of observing exactly x events or successes in a unit of time events or successes in a unit of time or space is given byor space is given by

.. μ is the mean number of occurrences μ is the mean number of occurrences

per unit time or space, that is, per unit time or space, that is, .. In a most amazing coincidence, it In a most amazing coincidence, it

turns out that turns out that !!

P( )!

xex

x

E( )X

2

A Fishing Trip TooA Fishing Trip Too Dr. A is fond of fly-fishing in the Colorado Dr. A is fond of fly-fishing in the Colorado

mountain streams. His long-run average is mountain streams. His long-run average is one catch per hourone catch per hour. On the last day of his . On the last day of his vacation, he stops to eat supper. He figures vacation, he stops to eat supper. He figures he will fish two more hours and then pack up. he will fish two more hours and then pack up. What is the probability he will catch at least What is the probability he will catch at least one more fish?one more fish?

Note: μ Note: μ = 2 (in a two hour period).= 2 (in a two hour period).

0 2

2

P(at least one) 1 P(0)

21

0!

1 .865

e

e

More Poisson FishingMore Poisson Fishing

What is the probability Dr. A will catch one more What is the probability Dr. A will catch one more fish?fish?

What is the probability Dr. A will catch more than What is the probability Dr. A will catch more than one more fish?one more fish?

1 222

P(1) 2 .2711!

ee

P( 1) P( 1) P( 1) .865 .271 .594X X X

Discrete DistributionsDiscrete Distributions The examples we have just seen are The examples we have just seen are

common common discretediscrete distributions, distributions, That is, the r.v. in question takes only That is, the r.v. in question takes only

discrete values (with P>0).discrete values (with P>0). We have also seen that discrete r.v.’s may We have also seen that discrete r.v.’s may

be be finitefinite or or infiniteinfinite with regard to the with regard to the number of values they can take (with number of values they can take (with P>0).P>0).

There are many more such discrete There are many more such discrete distributions, and we should mention some distributions, and we should mention some of them just so you are aware they are of them just so you are aware they are available.available.

The The multinomialmultinomial distribution is like the binomial, distribution is like the binomial, except that it has more than two categories with except that it has more than two categories with probabilities for each. The r.v. is not based on a probabilities for each. The r.v. is not based on a single value but a vector giving the counts of single value but a vector giving the counts of each possible outcome. The counts have to add each possible outcome. The counts have to add up to the number of trials. You could use this, for up to the number of trials. You could use this, for example, to calculate the probability of getting 2 example, to calculate the probability of getting 2 fives and 2 sixes in 4 tosses of a die, written as fives and 2 sixes in 4 tosses of a die, written as P(0,0,0,0,2,2).P(0,0,0,0,2,2).

1 21 2 1 2

1 2

!P( , ,..., )

! ! !kxx x

k kk

nx x x p p p

x x x

0 0 0 0 2 2

4 3

4! 1 1 1 1 1 1P(0,0,0,0,2,2)

0!0!0!0!2!2! 6 6 6 6 6 6

1 1 16

6 6 216

The The geometricgeometric distribution is used when you are distribution is used when you are interested in the number of trials until the first interested in the number of trials until the first success. This is an infinite-valued distribution like success. This is an infinite-valued distribution like Poisson. You could use this, for example, to Poisson. You could use this, for example, to calculate the probability that you find the first calculate the probability that you find the first defective on the 10defective on the 10thth trial, if the probability of a trial, if the probability of a defective is .1.defective is .1.

More often, we are interested in the More often, we are interested in the cumulativecumulative probability (finding the first defective by the 10probability (finding the first defective by the 10thth trial):trial):

1P( ) where (1 )xx pq q p

10

P( ) 1

P( 10) 1 .9 .65

xX x q

X

9P(10) (.1)(.9) .039

The The negative binomialnegative binomial distribution is an extension distribution is an extension of the geometric. Instead of counting the number of the geometric. Instead of counting the number of trials until the first success, we count the of trials until the first success, we count the number of failures, X, until we have r successes. number of failures, X, until we have r successes. For example, find the probability that you have to For example, find the probability that you have to examine 10 items to find 3 defectives, if the examine 10 items to find 3 defectives, if the probability of a defective is .1.probability of a defective is .1.

1P( , ) where (1 )

1r xx r

x r p q q pr

3 77 3 1P(7,3) (.1) (.9) .017

3 1