Download QMB 9e Chapter 3 - Probability Distributions

Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2004 Thomson/South-Western
Slide 1
Chapter 3, Part B
Continuous Probability Distributions



f (x)
Uniform Probability Distribution
Normal Probability Distribution
Exponential Probability Distribution
f (x) Exponential
Uniform
f (x)
Normal
x
x
x
© 2004 Thomson/South-Western
Slide 2
Continuous Probability Distributions



A continuous random variable can assume any value
in an interval on the real line or in a collection of
intervals.
It is not possible to talk about the probability of the
random variable assuming a particular value.
Instead, we talk about the probability of the random
variable assuming a value within a given interval.
© 2004 Thomson/South-Western
Slide 3
Continuous Probability Distributions

f (x)
The probability of the random variable assuming a
value within some given interval from x1 to x2 is
defined to be the area under the graph of the
probability density function between x1 and x2.
f (x) Exponential
Uniform
f (x)
x1 x 2
Normal
x1 xx12 x2
x
x1 x 2
© 2004 Thomson/South-Western
x
x
Slide 4
Uniform Probability Distribution


A random variable is uniformly distributed
whenever the probability is proportional to the
interval’s length.
The uniform probability density function is:
f (x) = 1/(b – a) for a < x < b
=0
elsewhere
where: a = smallest value the variable can assume
b = largest value the variable can assume
© 2004 Thomson/South-Western
Slide 5
Uniform Probability Distribution

Expected Value of x
E(x) = (a + b)/2

Variance of x
Var(x) = (b - a)2/12
© 2004 Thomson/South-Western
Slide 6
Example: Slater's Buffet

Uniform Probability Distribution
Slater customers are charged
for the amount of salad they take.
Sampling suggests that the
amount of salad taken is
uniformly distributed
between 5 ounces and 15 ounces.
© 2004 Thomson/South-Western
Slide 7
Example: Slater's Buffet

Uniform Probability Density Function
f(x) = 1/10 for 5 < x < 15
=0
elsewhere
where:
x = salad plate filling weight
© 2004 Thomson/South-Western
Slide 8
Example: Slater's Buffet

Expected Value of x
E(x) = (a + b)/2
= (5 + 15)/2
= 10

Variance of x
Var(x) = (b - a)2/12
= (15 – 5)2/12
= 8.33
© 2004 Thomson/South-Western
Slide 9
Example: Slater's Buffet

Uniform Probability Distribution
for Salad Plate Filling Weight
f(x)
1/10
5
10
15
Salad Weight (oz.)
© 2004 Thomson/South-Western
x
Slide 10
Example: Slater's Buffet
What is the probability that a customer
will take between 12 and 15 ounces of salad?
f(x)
P(12 < x < 15) = 1/10(3) = .3
1/10
5
10 12
15
Salad Weight (oz.)
© 2004 Thomson/South-Western
x
Slide 11
Normal Probability Distribution


The normal probability distribution is the most
important distribution for describing a continuous
random variable.
It is widely used in statistical inference.
© 2004 Thomson/South-Western
Slide 12
Normal Probability Distribution

It has been used in a wide variety of applications:
Heights
of people
© 2004 Thomson/South-Western
Scientific
measurements
Slide 13
Normal Probability Distribution

It has been used in a wide variety of applications:
Test
scores
© 2004 Thomson/South-Western
Amounts
of rainfall
Slide 14
Normal Probability Distribution

Normal Probability Density Function
1
 ( x   )2 /2 2
f (x) 
e
 2
where:
 = mean
 = standard deviation
 = 3.14159
e = 2.71828
© 2004 Thomson/South-Western
Slide 15
Normal Probability Distribution

Characteristics
The distribution is symmetric, and is bell-shaped.
x
© 2004 Thomson/South-Western
Slide 16
Normal Probability Distribution

Characteristics
The entire family of normal probability
distributions is defined by its mean  and its
standard deviation  .
Standard Deviation 
Mean 
© 2004 Thomson/South-Western
x
Slide 17
Normal Probability Distribution

Characteristics
The highest point on the normal curve is at the
mean, which is also the median and mode.
x
© 2004 Thomson/South-Western
Slide 18
Normal Probability Distribution

Characteristics
The mean can be any numerical value: negative,
zero, or positive.
x
-10
0
© 2004 Thomson/South-Western
20
Slide 19
Normal Probability Distribution

Characteristics
The standard deviation determines the width of the
curve: larger values result in wider, flatter curves.
 = 15
 = 25
x
© 2004 Thomson/South-Western
Slide 20
Normal Probability Distribution

Characteristics
Probabilities for the normal random variable are
given by areas under the curve. The total area
under the curve is 1 (.5 to the left of the mean and
.5 to the right).
.5
.5
x
© 2004 Thomson/South-Western
Slide 21
Normal Probability Distribution

Characteristics
68.26% of values of a normal random variable
are within +/- 1 standard deviation of its mean.
95.44% of values of a normal random variable
are within +/- 2 standard deviations of its mean.
99.72% of values of a normal random variable
are within +/- 3 standard deviations of its mean.
© 2004 Thomson/South-Western
Slide 22
Normal Probability Distribution

Characteristics
99.72%
95.44%
68.26%
 – 3
 – 1
 – 2
© 2004 Thomson/South-Western

 + 3
 + 1
 + 2
x
Slide 23
Standard Normal Probability Distribution
A random variable having a normal distribution
with a mean of 0 and a standard deviation of 1 is
said to have a standard normal probability
distribution.
© 2004 Thomson/South-Western
Slide 24
Standard Normal Probability Distribution
The letter z is used to designate the standard
normal random variable.
1
z
0
© 2004 Thomson/South-Western
Slide 25
Standard Normal Probability Distribution

Converting to the Standard Normal Distribution
z
x

We can think of z as a measure of the number of
standard deviations x is from .
© 2004 Thomson/South-Western
Slide 26
Using Excel to Compute
Standard Normal Probabilities

Excel has two functions for computing probabilities
and z values for a standard normal distribution:
NORM S DIST is used to compute the cumulative
NORMSDIST
probability given a z value.
NORMSINV
NORM S INV is used to compute the z value
given a cumulative probability.
(The “S” in the function names reminds
us that they relate to the standard
normal probability distribution.)
© 2004 Thomson/South-Western
Slide 27
Using Excel to Compute
Standard Normal Probabilities

Formula Worksheet
1
2
3
4
5
6
7
8
9
A
B
Probabilities: Standard Normal Distribution
P (z < 1.00)
P (0.00 < z < 1.00)
P (0.00 < z < 1.25)
P (-1.00 < z < 1.00)
P (z > 1.58)
P (z < -0.50)
© 2004 Thomson/South-Western
=NORMSDIST(1)
=NORMSDIST(1)-NORMSDIST(0)
=NORMSDIST(1.25)-NORMSDIST(0)
=NORMSDIST(1)-NORMSDIST(-1)
=1-NORMSDIST(1.58)
=NORMSDIST(-0.5)
Slide 28
Using Excel to Compute
Standard Normal Probabilities

Value Worksheet
1
2
3
4
5
6
7
8
9
A
B
Probabilities: Standard Normal Distribution
P (z < 1.00)
P (0.00 < z < 1.00)
P (0.00 < z < 1.25)
P (-1.00 < z < 1.00)
P (z > 1.58)
P (z < -0.50)
© 2004 Thomson/South-Western
0.8413
0.3413
0.3944
0.6827
0.0571
0.3085
Slide 29
Using Excel to Compute
Standard Normal Probabilities

Formula Worksheet
1
2
3
4
5
6
A
B
Finding z Values, Given Probabilities
z value with .10 in upper tail
z value with .025 in upper tail
z value with .025 in lower tail
© 2004 Thomson/South-Western
=NORMSINV(0.9)
=NORMSINV(0.975)
=NORMSINV(0.025)
Slide 30
Using Excel to Compute
Standard Normal Probabilities

Value Worksheet
1
2
3
4
5
6
A
B
Finding z Values, Given Probabilities
z value with .10 in upper tail
z value with .025 in upper tail
z value with .025 in lower tail
© 2004 Thomson/South-Western
1.28
1.96
-1.96
Slide 31
Example: Pep Zone

Standard Normal Probability Distribution
Pep Zone sells auto parts and supplies
including a popular multi-grade motor
oil. When the stock of this oil drops to
Pep
Zone
20 gallons, a replenishment order is
5w-20
placed.
Motor Oil
© 2004 Thomson/South-Western
Slide 32
Example: Pep Zone
Pep
Zone
5w-20
Motor Oil

Standard Normal Probability Distribution
The store manager is concerned that sales are
being lost due to stockouts while waiting for an
order. It has been determined that demand during
replenishment leadtime is normally distributed with
a mean of 15 gallons and a standard deviation of 6
gallons.
The manager would like to know the probability
of a stockout, P(x > 20).
© 2004 Thomson/South-Western
Slide 33
Example: Pep Zone
Pep
Zone

Solving for the Stockout Probability
5w-20
Motor Oil
Step 1: Convert x to the standard normal distribution.
z = (x - )/
= (20 - 15)/6
= .83
Step 2: Find the area under the standard normal
curve between the mean and z = .83.
see next slide
© 2004 Thomson/South-Western
Slide 34
Example: Pep Zone
Pep
Zone
5w-20
Motor Oil
Probability Table for the
Standard Normal Distribution

z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.
.
.
.
.
.
.
.
.
.
.
.5
.1915 .1695 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
.6
.2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549
.7
.2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
.8
.2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133
.9
.3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389
.
.
.
.
.
.
.
.
.
.
.
P(0 < z < .83)
© 2004 Thomson/South-Western
Slide 35
Example: Pep Zone
Pep
Zone

Solving for the Stockout Probability
5w-20
Motor Oil
Step 3: Compute the area under the standard normal
curve to the right of z = .83.
P(z > .83) = .5 – P(0 < z < .83)
= 1- .2967
= .2033
Probability
of a stockout
© 2004 Thomson/South-Western
P(x > 20)
Slide 36
Example: Pep Zone
Pep
Zone

5w-20
Motor Oil
Solving for the Stockout Probability
Area = .5 - .2967
Area = .2967
= .2033
0
© 2004 Thomson/South-Western
.83
z
Slide 37
Example: Pep Zone
Pep
Zone
5w-20
Motor Oil

Standard Normal Probability Distribution
If the manager of Pep Zone wants the probability
of a stockout to be no more than .05, what should the
reorder point be?
© 2004 Thomson/South-Western
Slide 38
Example: Pep Zone
Pep
Zone
5w-20
Motor Oil

Solving for the Reorder Point
Area = .4500
Area = .0500
0
© 2004 Thomson/South-Western
z.05
z
Slide 39
Example: Pep Zone
Pep
Zone

5w-20
Motor Oil
Solving for the Reorder Point
Step 1: Find the z-value that cuts off an area of .05
in the right tail of the standard normal
distribution.
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.
.
.
.
.
.
.
.
.
.
.
1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441
1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545
1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706
1.9 .4713 .4719 .4726 We
.4732look
.4738
.4750 .4756 .4761 .4767
up.4744
the area
.
.
.
.
. (.5 - ..05 = .45)
.
.
.
.
.
© 2004 Thomson/South-Western
Slide 40
Example: Pep Zone
Pep
Zone

Solving for the Reorder Point
5w-20
Motor Oil
Step 2: Convert z.05 to the corresponding value of x.
x =  + z.05
= 15 + 1.645(6)
= 24.87 or 25
A reorder point of 25 gallons will place the probability
of a stockout during leadtime at (slightly less than) .05.
© 2004 Thomson/South-Western
Slide 41
Example: Pep Zone
Pep
Zone
5w-20
Motor Oil

Solving for the Reorder Point
By raising the reorder point from 20 gallons to
25 gallons on hand, the probability of a stockout
decreases from about .20 to .05.
This is a significant decrease in the chance that Pep
Zone will be out of stock and unable to meet a
customer’s desire to make a purchase.
© 2004 Thomson/South-Western
Slide 42
Using Excel to Compute
Normal Probabilities

Excel has two functions for computing cumulative
probabilities and x values for any normal
distribution:
NORMDIST is used to compute the cumulative
probability given an x value.
NORMINV is used to compute the x value given
a cumulative probability.
© 2004 Thomson/South-Western
Slide 43
Using Excel to Compute
Normal Probabilities
Formula Worksheet

1
2
3
4
5
6
7
8
Pep
Zone
5w-20
Motor Oil
A
B
Probabilities: Normal Distribution
P (x > 20) =1-NORMDIST(20,15,6,TRUE)
Finding x Values, Given Probabilities
x value with .05 in upper tail =NORMINV(0.95,15,6)
© 2004 Thomson/South-Western
Slide 44
Using Excel to Compute
Normal Probabilities
5w-20
Motor Oil
Value Worksheet

1
2
3
4
5
6
7
8
Pep
Zone
A
B
Probabilities: Normal Distribution
P (x > 20) 0.2023
Finding x Values, Given Probabilities
x value with .05 in upper tail 24.87
Note: P(x > 20) = .2023 here using Excel, while our
previous manual approach using the z table yielded
.2033 due to our rounding of the z value.
© 2004 Thomson/South-Western
Slide 45
Exponential Probability Distribution


The exponential probability distribution is useful in
describing the time it takes to complete a task.
The exponential random variables can be used to
describe:
Time between
vehicle arrivals
at a toll booth
Time required
to complete
a questionnaire
© 2004 Thomson/South-Western
Distance between
major defects
in a highway
Slide 46
Exponential Probability Distribution

Density Function
f ( x) 
1

e  x /  for x > 0,  > 0
where:
 = mean
e = 2.71828
© 2004 Thomson/South-Western
Slide 47
Exponential Probability Distribution

Cumulative Probabilities
P ( x  x0 )  1  e  xo / 
where:
x0 = some specific value of x
© 2004 Thomson/South-Western
Slide 48
Using Excel to Compute
Exponential Probabilities
The EXPONDIST function can be used to compute
exponential probabilities.
The EXPONDIST function has three arguments:
1st
The value of the random variable x
2nd 1/m
3rd
the inverse of the mean
number of occurrences
“TRUE” or “FALSE”
in an interval
we will always enter
“TRUE” because we’re seeking a
cumulative probability
© 2004 Thomson/South-Western
Slide 49
Using Excel to Compute
Exponential Probabilities

Formula Worksheet
A
1
2
3 P (x < 18)
4 P (6 < x < 18)
5 P (x > 8)
6
B
Probabilities: Exponential Distribution
=EXPONDIST(18,1/15,TRUE)
=EXPONDIST(18,1/15,TRUE)-EXPONDIST(6,1/15,TRUE)
=1-EXPONDIST(8,1/15,TRUE)
© 2004 Thomson/South-Western
Slide 50
Using Excel to Compute
Exponential Probabilities

Value Worksheet
A
1
2
3 P (x < 18)
4 P (6 < x < 18)
5 P (x > 8)
6
B
Probabilities: Exponential Distribution
0.6988
0.3691
0.5866
© 2004 Thomson/South-Western
Slide 51
Example: Al’s Full-Service Pump

Exponential Probability Distribution
The time between arrivals of cars
at Al’s full-service gas pump follows
an exponential probability distribution
with a mean time between arrivals of
3 minutes. Al would like to know the
probability that the time between two successive
arrivals will be 2 minutes or less.
© 2004 Thomson/South-Western
Slide 52
Example: Al’s Full-Service Pump

Exponential Probability Distribution
f(x)
P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866
.4
.3
.2
.1
x
1
2
3
4
5
6
7
8
9 10
Time Between Successive Arrivals (mins.)
© 2004 Thomson/South-Western
Slide 53
Using Excel to Compute
Exponential Probabilities

Formula Worksheet
1
2
3
4
A
B
Probabilities: Exponential Distribution
P (x < 2)
=EXPONDIST(2,1/3,TRUE)
© 2004 Thomson/South-Western
Slide 54
Using Excel to Compute
Exponential Probabilities

Value Worksheet
1
2
3
4
A
B
Probabilities: Exponential Distribution
P (x < 2)
0.4866
© 2004 Thomson/South-Western
Slide 55
Relationship between the Poisson
and Exponential Distributions
The Poisson distribution
provides an appropriate description
of the number of occurrences
per interval
The exponential distribution
provides an appropriate description
of the length of the interval
between occurrences
© 2004 Thomson/South-Western
Slide 56
End of Chapter 3, Part B
© 2004 Thomson/South-Western
Slide 57