Doane Chapter 07
-
Upload
thomasmcarter -
Category
Documents
-
view
220 -
download
0
Transcript of Doane Chapter 07
-
8/22/2019 Doane Chapter 07
1/67
-
8/22/2019 Doane Chapter 07
2/67
Continuous Distributions
Chapter
7
Continuous Variables
Describing a Continuous Distribution
Uniform Continuous Distribution
Normal Distribution
Standard Normal Distribution
Normal Approximation to the Binomial (Optional)
Normal Approximation to the Poisson (Optional)
Exponential Distribution
Triangular Distribution
-
8/22/2019 Doane Chapter 07
3/67
Continuous Variables
Discrete Variable each value ofXhas its own
probability P(X).
Continuous Variable events are intervals andprobabilities are areas underneath smooth
curves. A single point has no probability.
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Events as In tervals
-
8/22/2019 Doane Chapter 07
4/67
Describing a Continuous Distribution
Probability Density Function (PDF)
For a continuous
random variable,the PDF is an
equation that shows
the height of the
curve f(x) at eachpossible value ofX
over the range ofX.
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
PDFs and CDFs
Normal PDF
-
8/22/2019 Doane Chapter 07
5/67
Describing a Continuous Distribution
Continuous PDFs:
Denoted f(x)
Must be nonnegative Total area under
curve = 1
Mean, variance and
shape depend onthe PDFparameters
Reveals the shapeof the distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
PDFs and CDFs
Normal PDF
-
8/22/2019 Doane Chapter 07
6/67
Describing a Continuous Distribution
Continuous CDFs:
Denoted F(x)
Shows P(X
-
8/22/2019 Doane Chapter 07
7/67
Describing a Continuous Distribution
Continuous probability functions are smooth curves.
Unlike discrete
distributions, thearea at any
single point = 0.
The entire area under
any PDF must be 1. Mean is the balance
point of the distribution.
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Probabi l i t ies as Areas
-
8/22/2019 Doane Chapter 07
8/67
Describing a Continuous Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Expected Value and Variance
-
8/22/2019 Doane Chapter 07
9/67
Uniform Continuous Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Uni form Distr ibu t ion
IfXis a random variable that is uniformly
distributed between a and b, its PDF has
constant height. Denoted U(a,b)
Area =
base x height =
(b-a) x 1/(b-a) = 1
-
8/22/2019 Doane Chapter 07
10/67
Uniform Continuous Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Uni form Distr ibu t ion
-
8/22/2019 Doane Chapter 07
11/67
Uniform Continuous Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Uni form Distr ibu t ion
The CDF increases linearly to 1.
CDF formula is
(x-a)/(b-a)
-
8/22/2019 Doane Chapter 07
12/67
Uniform Continuous Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example: Anesthesia Effect iveness
An oral surgeon injects a painkiller prior to
extracting a tooth. Given the varying
characteristics of patients, the dentist views the
time for anesthesia effectiveness as a uniform
random variable that takes between 15 minutes
and 30 minutes.
Xis U(15, 30) a = 15, b = 30, find the mean and standard
deviation.
-
8/22/2019 Doane Chapter 07
13/67
Uniform Continuous Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example: Anesthesia Effect iveness
m =a + b
2=
15 + 30
2= 22.5 minutes
s =(ba)2
12= 4.33 minutes(30 15)
2
12=
Find the probability that the anesthetic takes between
20 and 25 minutes.P(c
-
8/22/2019 Doane Chapter 07
14/67
Uniform Continuous Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example: Anesthesia Effect iveness
P(20
-
8/22/2019 Doane Chapter 07
15/67
Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Normal Distr ibu t ion
Normal or Gaussian distribution was named for
German mathematician Karl Gauss (1777
1855).
Defined by two parameters, m and s
Denoted N(m, s)
Domain is
-
8/22/2019 Doane Chapter 07
16/67
Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Normal Distr ibu t ion
-
8/22/2019 Doane Chapter 07
17/67
Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Normal Distr ibu t ion
Normal PDF f(x) reaches a maximum at m and
has points of inflection at m + s
Bell-shaped curve
-
8/22/2019 Doane Chapter 07
18/67
Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Normal Distr ibu t ion
Normal CDF
-
8/22/2019 Doane Chapter 07
19/67
Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Normal Distr ibu t ion
All normal distributions have the same shape but
differ in the axis scales.
Diameters of golf balls
m = 42.70mms = 0.01mm
CPA Exam Scores
m = 70s = 10
-
8/22/2019 Doane Chapter 07
20/67
Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
What is Normal?
A normal random variable should:
Be measured on a continuous scale.
Possess clear central tendency. Have only one peak (unimodal).
Exhibit tapering tails.
Be symmetric about the mean (equal tails).
-
8/22/2019 Doane Chapter 07
21/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics o f the Standard Normal
Since for every value ofm and s, there is a
different normal distribution, we transform a
normal random variable to a standard normal
distribution with m = 0 and s = 1 using the
formula:
z=xm
s Denoted N(0,1)
-
8/22/2019 Doane Chapter 07
22/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics o f the Standard Normal
-
8/22/2019 Doane Chapter 07
23/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics o f the Standard Normal Standard normal PDF f(x) reaches a maximum at
0 and has points of inflection at +1.
Shape is
unaffected by the
transformation.
It is still a bell-
shaped curve.
-
8/22/2019 Doane Chapter 07
24/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics o f the Standard Normal Standard normal CDF
-
8/22/2019 Doane Chapter 07
25/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics o f the Standard Normal A common scale from -3 to +3 is used.
Entire area under the curve is unity.
The probability of an event P(z1 < Z< z2) is adefinite integral off(z).
However, standard normal tables or Excel
functions can be used to find the desired
probabilities.
-
8/22/2019 Doane Chapter 07
26/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Normal A reas from Append ix C-1 Appendix C-1 allows you to find the area under
the curve from 0 to z.
For example, findP(0 < Z< 1.96):
-
8/22/2019 Doane Chapter 07
27/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
-
8/22/2019 Doane Chapter 07
28/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Normal A reas from Append ix C-1
.5000
.5000 - .4750 = .0250
Now find P(Z< 1.96):
-
8/22/2019 Doane Chapter 07
29/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc All rights reserved
Normal A reas from Append ix C-1 Now find P(-1.96 < Z< 1.96).
Due to symmetry, P(-1.96 < Z) is the same as
P(Z< 1.96).
So, P(-1.96 < Z< 1.96) = .4750 + .4750 = .9500
or 95% of the area under the curve.
.9500
-
8/22/2019 Doane Chapter 07
30/67
Standard Normal Distribution
McGraw-Hill/Irwin
2007 The McGraw-Hill Companies Inc All rights reserved
Basis fo r the Emp ir ical Rule Approximately 95% of the area under the curve
is between + 2s
Approximately 99.7% of the area under the curve
is between + 3s
-
8/22/2019 Doane Chapter 07
31/67
Standard Normal Distribution
McGraw-Hill/Irwin
2007 The McGraw-Hill Companies Inc All rights reserved
Normal A reas from Append ix C-2 Appendix C-2 allows you to find the area under
the curve from the left ofz(similar to Excel).
For example,
.9500
P(Z< -1.96)P(Z< 1.96) P(-1.96 < Z< 1.96)
-
8/22/2019 Doane Chapter 07
32/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
-
8/22/2019 Doane Chapter 07
33/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Normal Areas from Append ices C-1 or C-2 Appendices C-1 and C-2 yield identical results.
Use whichever table is easiest.
Findingz for a Given Area Appendices C-1 and C-2 be used to find the
z-value corresponding to a given probability.
For example, what z-value defines the top 1% of
a normal distribution?
This implies that 49% of the area lies between 0
and z.
-
8/22/2019 Doane Chapter 07
34/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Look for an
area of .4900
in Appendix
C-1:
Findingz for a Given Area
Without
interpolation,
the closest wecan get is
z = 2.33
-
8/22/2019 Doane Chapter 07
35/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Some important Normal areas:
Findingz for a Given Area
-
8/22/2019 Doane Chapter 07
36/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Finding Normal A reas w ith Excel
-
8/22/2019 Doane Chapter 07
37/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Finding Normal A reas w ith Excel
d d l b
-
8/22/2019 Doane Chapter 07
38/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Finding Normal A reas w ith Excel
S d d l b
-
8/22/2019 Doane Chapter 07
39/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Finding Normal A reas w ith Excel
S d d N l Di ib i
-
8/22/2019 Doane Chapter 07
40/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Find ing A reas by us ing Standardized Variables
Suppose John took an economics exam and
scored 86 points. The class mean was 75 with a
standard deviation of 7. What percentile is John
in (i.e., find P(X< 86)?
zJohn =xm
s=
86 75
7= 11/7 = 1.57
So Johns score is 1.57 standard deviations about
the mean.
S d d N l Di ib i
-
8/22/2019 Doane Chapter 07
41/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Find ing A reas by us ing Standardized Variables
P(X< 86) = P(Z< 1.57) = .9418
(from Appendix C-2)
So, John is approximately in the 94th percentile.
St d d N l Di t ib ti
-
8/22/2019 Doane Chapter 07
42/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Inverse Normal You can manipulate the transformation formula to
find the normal percentile values (e.g., 5th, 10th,
25th, etc.): x= m + zs Here are some common percentiles
St d d N l Di t ib ti
-
8/22/2019 Doane Chapter 07
43/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Using Excel Without Standard izing Excels NORMDIST and NORMINV function allow
you to evaluate areas without standardizing.
For example, let m = 2.040 cm and s = .001 cm,
what is the probability that a given steel bearingwill have a diameter between 2.039 and 2.042cm?
In other words, P(2.039
-
8/22/2019 Doane Chapter 07
44/67
Standard Normal Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Using Excel Without Standardizing
P(X < 2.042) = .9773 P(X < 2.039) = .1587
P(2.039
-
8/22/2019 Doane Chapter 07
45/67
Normal Approximation to the Binomial
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
When is App rox imat ion Needed? Binomial probabilities are difficult to calculate
when n is large.
Use a normal approximation to the binomial.
As n becomes large, the binomial bars become
more continuous and smooth.
N l A i ti t th Bi i l
-
8/22/2019 Doane Chapter 07
46/67
Normal Approximation to the Binomial
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
When is App rox imat ion Needed?
Rule of thumb: when np > 10 and n(1-p) > 10,
then it is appropriate to use the normal
approximation to the binomial. In this case, the binomial mean and standard
deviation will be equal to the normal m and s,
respectively.
m = nps = np(1-p)
Normal Approximation to the Binomial
-
8/22/2019 Doane Chapter 07
47/67
Normal Approximation to the Binomial
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example Coin Flips If we were to flip a coin n = 32 times and p = .50,
are the requirements for a normal approximation
to the binomial met?
Are np > 5 and n(1-p) > 10? np = 32 x .50 = 16
n(1-p) = 32 x (1 - .50) = 16
So, a normal approximation can be used.
When translating a discrete scale into a
continuous scale, care must be taken about
individual points.
Normal Approximation to the Binomial
-
8/22/2019 Doane Chapter 07
48/67
Normal Approximation to the Binomial
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example Coin Flips For example, find the probability of more than 17
heads in 32 flips of a fair coin.
This can be written
as P(X> 18). However, more
than 17 actually
falls between 17
and 18 on a discretescale.
Normal Approximation to the Binomial
-
8/22/2019 Doane Chapter 07
49/67
Normal Approximation to the Binomial
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example Coin Flips Since the cutoff point for more than 17 is halfway
between 17 and 18, we add 0.5 to the lower limit
and find P(X> 17.5).
This addition toXis called the Continuity
Correction.
At this point, the problem can be completed as
any normal distribution problem.
Normal Approximation to the Binomial
-
8/22/2019 Doane Chapter 07
50/67
Normal Approximation to the Binomial
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Cont inu i ty Correct ion The table below shows some events and their
cutoff point for the normal approximation.
Normal Approximation to the Poisson
-
8/22/2019 Doane Chapter 07
51/67
Normal Approximation to the Poisson
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
When is App rox imat ion Needed? The normal approximation to the Poisson works
best when lis large (e.g., when l exceeds thevalues in Appendix B).
Set the normal m and s equal to the Poisson mean
and standard deviation.
m = ls = l
Normal Approximation to the Poisson
-
8/22/2019 Doane Chapter 07
52/67
Normal Approximation to the Poisson
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example Util i ty B i l ls On Wednesday between 10A.M. and noon
customer billing inquiries arrive at a mean rate of
42 inquiries per hour at Consumers Energy. What
is the probability of receiving more than 50 calls?
l = 42 which is too big to use the Poisson table. Use the normal approximation with
m = l = 42s = l = 42 = 6.48074
Normal Approximation to the Poisson
-
8/22/2019 Doane Chapter 07
53/67
Normal Approximation to the Poisson
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example Util i ty B i l ls
To find P(X> 50) calls, use the continuity-
corrected cutoff point halfway between 50 and 51
(i.e.,X= 50.5).
At this point, the problem can be completed as
any normal distribution problem.
Exponential Distribution
-
8/22/2019 Doane Chapter 07
54/67
Exponential Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Exponent ia l Distr ibut ion If events per unit of time follow a Poisson
distribution, the waiting time until the next event
follows the Exponential distribution.
Waiting time until the next event is a continuousvariable.
Exponential Distribution
-
8/22/2019 Doane Chapter 07
55/67
Exponential Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Exponent ia l Distr ibut ion
Exponential Distribution
-
8/22/2019 Doane Chapter 07
56/67
Exponential Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Exponent ia l Distr ibut ion
Probability of waiting more thanx Probability of waiting less thanx
Exponential Distribution
-
8/22/2019 Doane Chapter 07
57/67
Exponential Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example Cus tomer Wait ing Time Between 2P.M. and 4P.M. on Wednesday, patient
insurance inquiries arrive at Blue Choice
insurance at a mean rate of 2.2 calls per minute.
What is the probability of waiting more than 30seconds (i.e., 0.50 minutes) for the next call?
Set l = 2.2 events/min andx= 0.50 min P(X> 0.50) = e
lx= e
(2.2)(0.5)
= .3329or 33.29% chance of waiting more than 30
seconds for the next call.
Exponential Distribution
-
8/22/2019 Doane Chapter 07
58/67
Exponential Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Example Cus tomer Wait ing Time
P(X> 0.50) P(X< 0.50)
Exponential Distribution
-
8/22/2019 Doane Chapter 07
59/67
Exponential Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Inverse Exponential If the mean arrival rate is 2.2 calls per minute, we
want the 90th percentile for waiting time (the top
10% of waiting time).
Find thex-valuethat defines the
upper 10%.
Exponential Distribution
-
8/22/2019 Doane Chapter 07
60/67
Exponential Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Inverse Exponential P(Xx) = .10
So, elx = .10 -lx= ln(.10)
= -2.302585
x= 2.302585/l= 2.302585/2.2
= 1.0466 min. 90% of the calls will arrive within 1.0466 minutes
(62.8 seconds).
Exponential Distribution
-
8/22/2019 Doane Chapter 07
61/67
Exponential Distribution
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Inverse Exponential Quartiles for Exponential with l = 2.2
Exponential Distribution
-
8/22/2019 Doane Chapter 07
62/67
p
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Mean Time Between Events Exponential waiting times are described as
Mean time between events (MTBE) = 1/l 1/MTBE = l = mean events per unit of time In a hospital, if an event is patient arrivals in an
ER, and the MTBE is 20 minutes, thenl = 1/20 = 0.05 arrivals per minute (or 3/hour).
Exponential Distribution
-
8/22/2019 Doane Chapter 07
63/67
p
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Using Excel In Excel, use =EXPONDIST(x,l,1) to return the
left-tail area P(X
-
8/22/2019 Doane Chapter 07
64/67
g
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Triangular Distr ibut ion A simple distribution that can be symmetric or
skewed.
Ranges from a to band has a mode or peak at c
Denoted T(a,b,c)
Triangular Distribution
-
8/22/2019 Doane Chapter 07
65/67
g
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Character ist ics of the Triangular Distr ibu t ion
Triangular Distribution
-
8/22/2019 Doane Chapter 07
66/67
g
McGraw-Hill/Irwin 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Special Cases: Symmetric Triangu lar A symmetric triangular distribution is centered at 0
Lower limit is identical to the upper limit except for
the sign, with mode 0.
Mean m = 0, standard deviation s = b/ 6 =2.45
This distribution closely
resembles a standard
normal distribution N(0,1) Generate random triangular
data in Excel by summing
RAND()+RAND()
-
8/22/2019 Doane Chapter 07
67/67
Applied Statistics inBusiness and Economics
End of Chapter 7