bcor 1020 business statistics lecture 10 – february 19, 2008

BCOR 1020Business Statistics

Lecture 10 – February 19, 2008

Overview

• Chapter 6 – Discrete Distributions– Bernoulli Distribution– Binomial Distribution

Chapter 6 – Bernoulli Distribution

Bernoulli Experiments:• A Bernoulli Experiment is a random experiment, the

outcome of which can be classified in one of two mutually exclusive and exhaustive ways:– One outcome is arbitrarily labeled a “success” (denoted

X = 1) and the other a “failure” (denoted X = 0).

P(X = 1) = P(success) = P(X = 0) = P(failure) = 1 –

• Note that P(0) + P(1) = (1 – ) + = 1 and 0 < < 1.

• “Success” is usually defined as the less likely outcome so that < .5 for convenience.


Bernoulli Experiment Possible Outcomes Probability of “Success”

Flip a coin 1 = heads0 = tails

= .50

Some examples of Bernoulli experiments:

Inspect a jet turbine blade 1 = crack found0 = no crack found

= .001

Purchase a tank of gas 1 = pay by credit card0 = do not pay by credit card

= .78

Purchase a Lottery ticket 1 = all 6 numbers match0 = fewer than 6 match

= .0000002

Bernoulli Experiments:


The PDF of the Bernoulli Experiment:

1,0,)1()()( 1 xxfxXP xx

The Mean and Variance of the Bernoulli Experiment:

1)1(0)()(1

0x

xfxXE

221

0

22 )1()1()0()()()(x

xfxXV

22 )1()1()(

)1()1(

)1(2 / /

Chapter 6 – Binomial Distribution

Bernoulli Trials:• If we repeat a Bernoulli Experiment several (n)

times independently, we have n Bernoulli trials.– n Independent Bernoulli experiments– P(“success”) = on each of the n trials

• Example: Suppose we roll a single die 4 times and observe whether a “1” is rolled each time.

• This would constitute a set of n = 4 Bernoulli trials with …

= P(“success”) = P(“1” is rolled)

61


The Binomial Distribution:• If X denotes the number of “successes” observed in n

Bernoulli trials, then we say that X has the Binomial distribution with parameters n and .

• This is often denoted X~b(n,)

• We can determine the following about X based on the values of n and :

– The PDF of X, f(x) = P(X = x)

– The mean of X, = E(X)

– The variance of X, = V(X)– The std. deviation of X,

Chapter 6 – Binomial DistributionThe PDF of the Binomial Distribution:• To find the PMF of the Binomial distribution, we observe

that it consists of n independent Bernoulli distributions, each with parameter .

xnxxnx xPXP )1()0()1(

• The probability of observing x “successes” in n Bernoulli trials will be given by the product of …– P(x “successes”) and P((n – x) “failures”)

– And the number of ways in which this combination of successes and failures can occur

)!(!!xnx

nnCx


nx

CxfxXP xnxxnx

nxnxxn

,,2,1,0

,)1()1()()( )!(!!

nXE )(

)1()(2 nXV

The PDF of the Binomial Distribution:

The mean of the Binomial Distribution:

The variance and standard deviation of the Binomial Distribution:

)1( n


Parameters n = number of trials = probability of success

PDF

Excel function =BINOMDIST(k,n,,0)

Range X = 0, 1, 2, . . ., n

Mean n

Std. Dev.

Random data generation in Excel

Sum n values of =1+INT(2*RAND()) or use Excel’s Tools | Data Analysis

Comments Skewed right if < .50, skewed left if > .50, and symmetric if = .50.

!( ) (1 )

!( )!x n xn

P xx n x

(1 )n


• It is important to quick oil change shops to ensure that a car’s service time is not considered “late” by the customer.

• Service times are defined as either late or not late.

• X is the number of cars that are late out of the total number of cars serviced.

• Assumptions: - cars are independent of each other - probability of a late car is consistent

Example: Quick Oil Change Shop


1)1.0(10 n

949.0)9.0()1.0(10)1( n

xnxxnCxfxXP )1()()(

• Assuming that the probability that a car is late is P(late) = = 0.10,…

Example: Quick Oil Change Shop

a) For the next n = 10 cars serviced, what are the mean and standard deviation for the number of late cars?

b) What is the probability that exactly 2 of the next n = 10 cars serviced are late (i.e. P(X = 2))?

82210 )9.0()1.0()2()2( CfXP

1937.0)9.0()1.0(45)9.0()1.0( 8282!8!2!10

Clickers

A real estate agent estimates that her probability of selling a house is 10%. If she shows houses to 5 customers today, what is the expected number of houses she will sell?

A = 0.0

B = 0.2

C = 0.5

D = 1.0

Clickers

A real estate agent estimates that her probability of selling a house is 10%. If she shows houses to 5 customers today, what is the standard deviation for the number of houses she will sell?

A = 0.45

B = 0.67

C = 0.90

D = 0.95

Clickers

A real estate agent estimates that her probability of selling a house is 10%. If she shows houses to 5 customers today, what is the probability that she will sell one house?

A = 0.4095

B = 0.3281

C = 0.6719

D = 0.5905


Binomial Shape:• A binomial distribution is

1) skewed right if < .50,

2) skewed left if > .50,

3) and symmetric if = .50

• Skewness decreases as n increases, regardless of the value of .– For large n, the binomial distribution is symmetric!

• To illustrate, consider the following graphs:


p = .20Skewed right

p = .50Symmetric

p = .80Skewed left

n = 5

n = 10

n = 200.00

0.05

0.10

0.15

0.20

0.25

0 5 10 15 20

Number of Successes

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 5 10 15 20

Number of Successes

0.00

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0.25

0 5 10 15 20

Number of Successes

0.00

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0.35

0 5 10 15 20

Number of Successes

0.00

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Number of Successes

0.00

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0 5 10 15 20

Number of Successes

0.00

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0 5 10 15 20

Number of Successes

0.00

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Number of Successes

0.00

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0.45

0 5 10 15 20

Number of Successes


• Individual probabilities can be added to obtain any desired event probability.

• For example, the probability that a sample of 4 patients will contain at least 2 uninsured patients is

• HINT: What inequality means “at least?”

P(X 2) = P(2) + P(3) + P(4)

Compound Events:

In this example we can also use the complimentary probability, P(A’) = 1 – P(A):

P(X 2) = 1 – P(X < 2) = 1 – P(0) – P(1)


Compound Events:• We must determine which inequality corresponds to

our problem:

""

• “fewer than” = “less than” =

• “at most” = “no more than” =

• “more than” = “greater than” =

• “at least” = “no fewer than” =

""

""

""

Clickers

A real estate agent estimates that her probability of selling a house is 10%. If she shows houses to 5 customers today, what is the probability that she will sell at least one house?

A = 0.4095

B = 0.3281

C = 0.6719

D = 0.5905

Chapter 6 – Binomial DistributionRecognizing Binomial Applications:• Look for n independent Bernoulli trials with

constant probability of success.– Recall: A Bernoulli Experiment is one in which there

are two mutually exclusive and exhaustive outcomes (“success” or “failure”).

P(“success”) =

– If we repeat a Bernoulli Experiment n times independently then we have n Bernoulli trials with

P(“success”) = .

• If we are counting the number of “successes” observed in n Bernoulli trials, then we have a Binomial Distribution, X~b(n,).

ClickersIn a manufacturing process involving 5 mm bolts, our supplier guarantees that 95% of the bolts they sell to us meet specifications. If we select 50 of these bolts at random and observe the number of bolts that don’t meet specification, what type of probability distribution best describes our experiment?

A = Bernoulli Distribution

B = Binomial with n = 50, = 0.95

C = Binomial with n = 50, = 0.05

D = None of the above

bcor 1020 business statistics lecture 10 – february 19, 2008

Documents

binomial distribution

n bernoulli trials

bernoulli experiment

n independent bernoulli

n increases

large n

skewed left n

p0 p1 slide