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Chapter 4: Discrete Random Variables

Statistics

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McClave, Statistics, 11th ed. Chapter 4: DiscreteRandom Variables 2

Where Weve Been

Using probability to make inferencesabout populations

Measuring the reliability of theinferences

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McClave, Statistics, 11th ed. Chapter 4: DiscreteRandom Variables 3

Where Were Going

Develop the notion of a randomvariable

Numerical data and discrete randomvariables

Discrete random variables and their

probabilities

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4.1: Two Types of RandomVariables

A random variableis a variable hatassumes numerical values associated

with the random outcome of anexperiment, where one (and only one)numerical value is assigned to each

sample point.

4McClave, Statistics, 11th ed. Chapter 4:Discrete Random Variables

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A discreterandom variablecan assume acountable number of values. Number of steps to the top of the Eiffel Tower*

A continuousrandom variablecanassume any value along a given interval ofa number line.

The time a tourist stays at the toponce s/he gets there

*Believe it or not, the answer ranges from 1,652 to 1,789. See Great Buildings

http://www.greatbuildings.com/cgi-bin/glk?http://www.endex.com/gf/buildings/eiffel/eiffel.htmlhttp://www.greatbuildings.com/cgi-bin/glk?http://www.endex.com/gf/buildings/eiffel/eiffel.html

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Discreterandom variables Number of sales Number of calls Shares of stock People in line Mistakes per page

Continuousrandomvariables

Length Depth Volume Time Weight


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4.2: Probability Distributionsfor Discrete Random Variables

The probability distribution of adiscrete random variable is a graph,

table or formula that specifies theprobability associated with eachpossible outcome the random variablecan assume.

p(x) 0 for all values ofx p(x)= 1


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4.2: Probability Distributionsfor Discrete Random Variables

Say a random variablexfollows this pattern:

p(x)= (.3)(.7)x-1

forx> 0.

This table gives theprobabilities (rounded

to two digits) forxbetween 1 and 10.

x P(x)

1 .30

2 .21

3 .15

4 .11

5 .07

6 .05

7 .04

8 .02

9 .02

10 .01


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4.3: Expected Values ofDiscrete Random Variables

The mean, orexpected value,of adiscrete random variableis

( ) ( ).E x xp x


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The variance of adiscrete randomvariablexis

The standard deviation of a discrete

random variablex is

2 2 2[( ) ] ( ) ( ).E x x p x

2 2 2[( ) ] ( ) ( ).E x x p x


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)33(

)22(

)(

xP

xP

xP

ChebyshevsRule Empirical Rule

0 .68

.75 .95

.89 1.00



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In a roulette wheel in a U.S. casino, a $1 bet oneven wins $1 if the ball falls on an even number(same for odd, or red, or black).

The odds of winning this bet are 47.37%

9986.0526.5263.1$4737.1$

5263.)1$(

4737.)1$(

loseP

winP

On average, bettors lose about a nickel for each dollar they put down on a bet like this.(These are the bestbets for patrons.)

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4.4: The Binomial Distribution

A Binomial Random Variable

nidentical trials

Two outcomes: Success or Failure P(S) =p; P(F) = q= 1p

Trials are independent

xis the number of Successes in n trials


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A Binomial RandomVariable nidentical trials

Two outcomes: Successor Failure

P(S) =p; P(F) = q= 1p

Trials are independent

xis the number of Ss in ntrials

Flip a coin 3 times

Outcomes are Heads or Tails

P(H) = .5; P(F) = 1-.5 = .5

A head on flip idoesnt

change P(H) of flip i+ 1


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Results of 3 flips Probability Combined Summary

HHH (p)(p)(p) p3 (1)p3q0

HHT (p)(p)(q) p2

qHTH (p)(q)(p) p2q (3)p2q1

THH (q)(p)(p) p2q

HTT (p)(q)(q) pq2

THT (q)(p)(q) pq2 (3)p1q2

TTH (q)(q)(p) pq2

TTT (q)(q)(q) q3 (1)p0q315McClave, Statistics, 11th ed. Chapter 4:

Discrete Random Variables

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The Binomial Probability Distribution

p= P(S) on a single trial

q= 1p n= number of trials

x= number of successes

xnxqpxnxP

)(


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The Binomial Probability Distribution

xnxqpxnxP

)(


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Say 40% of theclass is female.

What is theprobability that 6of the first 10

students walkingin will be female?


1115.)1296)(.004096(.210

)6)(.4(.6

10

)(

6106

xnxqp

x

nxP


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MeanVariance

Standard Deviation

A Binomial Random Variable has

McClave, Statistics, 11th ed. Chapter 4:Discrete Random Variables

19

2npnpq

npq

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16250

2505.5.1000

5005.1000

2

npq

npq

np

For 1,000 coin flips,

The actual probability of getting exactly 500 heads out of 1000 flips isjust over 2.5%, but the probability of getting between 484 and 516 heads

(that is, within one standard deviation of the mean) is about 68%.


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4.5: The Poisson Distribution

Evaluates the probability of a (usuallysmall) number of occurrences out of many

opportunities in a Period of time

Area

Volume

Weight Distance

Other units of measurement


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!)(

x

exP

x

= mean number of occurrences in thegiven unit of time, area, volume, etc.

e = 2.71828.

=

2=22McClave, Statistics, 11th ed. Chapter 4:

Discrete Random Variables

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1008.!5

3

!)5(

35

e

x

e

xP

x

Say in a given stream there are an averageof 3 striped trout per 100 yards. What is theprobability of seeing 5 striped trout in the

next 100 yards, assuming a Poissondistribution?


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0141.!5

5.1

!)5(

5.15

e

x

e

xP

x

How about in the next 50 yards, assuming aPoisson distribution?

Since the distance is only half as long, is only

half as large.


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4.6: The HypergeometricDistribution

In the binomial situation, each trial wasindependent.

Drawing cards from a deck and replacingthe drawn card each time

If the card is notreplaced, each trial

depends on the previous trial(s). The hypergeometric distribution can be

used in this case.


25

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26

Randomly draw nelements from a setof Nelements, without replacement.

Assume there are rsuccesses and N-rfailures in the Nelements.

The hypergeometric random variable

is the number of successes,x, drawnfrom the ravailable in the nselections.

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27

nN

xn

rN

x

r

xP )(

where

N= the total number of elementsr = number of successes in the Nelementsn= number of elements drawn

X= the number of successes in the n elements

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28

nN

xn

rN

x

r

xP )(

)1(

)()(2

2

NN

nNnrNr

N

nr

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44.22.2)2()2()2or2(

22.

45

)1)(10(

210

22

510

2

5

)2()2(

FPMPFMP

FPMP

Suppose a customer at a pet store wants to buy two hamstersfor his daughter, but he wants two males or two females (i.e.,he wants only two hamsters in a few months)

If there are ten hamsters, five male and five female, what is the

probability of drawing two of the same sex? (With hamsters,its virtually a random selection.)

McClave, Statistics, 11th ed. Chapter 4:

Discrete Random Variables29

chapter 04

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