Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2003 South-Western/Thomson Learning™
Slide 1
Chapter 6
Continuous Probability Distributions



Uniform Probability Distribution
Normal Probability Distribution
Exponential Probability Distribution
f(x)

© 2003 South-Western/Thomson Learning™
x
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.
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.
© 2003 South-Western/Thomson Learning™
Slide 3
Uniform Probability Distribution


A random variable is uniformly distributed
whenever the probability is proportional to the
interval’s length.
Uniform Probability Density Function
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
© 2003 South-Western/Thomson Learning™
Slide 4
Uniform Probability Distribution

Expected Value of x
E(x) = (a + b)/2

Variance of x
Var(x) = (b - a)2/12
where: a = smallest value the variable can assume
b = largest value the variable can assume
© 2003 South-Western/Thomson Learning™
Slide 5
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.
The probability density function is
f(x) = 1/10 for 5 < x < 15
=0
elsewhere
where:
x = salad plate filling weight
© 2003 South-Western/Thomson Learning™
Slide 6
Example: Slater's Buffet

Uniform Probability Distribution
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
x
5
10 12
15
Salad Weight (oz.)
© 2003 South-Western/Thomson Learning™
Slide 7
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
© 2003 South-Western/Thomson Learning™
Slide 8
Normal Probability Distribution

Graph of the Normal Probability Density Function
f(x)

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x
Slide 9
Normal Probability Distribution

Characteristics of the Normal Probability
Distribution
• The shape of the normal curve is often illustrated
as a bell-shaped curve.
• Two parameters,  (mean) and s (standard
deviation), determine the location and shape of
the distribution.
• The highest point on the normal curve is at the
mean, which is also the median and mode.
• The mean can be any numerical value: negative,
zero, or positive.
… continued
© 2003 South-Western/Thomson Learning™
Slide 10
Normal Probability Distribution

Characteristics of the Normal Probability
Distribution
• The normal curve is symmetric.
• The standard deviation determines the width of
the curve: larger values result in wider, flatter
curves.
• The total area under the curve is 1 (.5 to the left of
the mean and .5 to the right).
• Probabilities for the normal random variable are
given by areas under the curve.
© 2003 South-Western/Thomson Learning™
Slide 11
Normal Probability Distribution

Percent (%) of Values in Some Commonly Used
Intervals
• 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.
© 2003 South-Western/Thomson Learning™
Slide 12
Normal Probability Distribution

Normal Probability Density Function
1
 ( x   ) 2 / 2s 2
f ( x) 
e
2 s
where:
 = mean
s = standard deviation
 = 3.14159
e = 2.71828
© 2003 South-Western/Thomson Learning™
Slide 13
Standard Normal Probability Distribution



A random variable that has a normal distribution
with a mean of zero and a standard deviation of one
is said to have a standard normal probability
distribution.
The letter z is commonly used to designate this
normal random variable.
Converting to the Standard Normal Distribution
z

x
s
We can think of z as a measure of the number of
standard deviations x is from .
© 2003 South-Western/Thomson Learning™
Slide 14
Using Excel to Compute
Standard Normal Probabilities

Excel has two functions for computing probabilities
and z values for a standard normal distribution:
• NORMSDIST is used to compute the cumulative
probability given a z value.
• NORMSINV is used to compute the z value given
a cumulative probability.
(The letter S in the above function names reminds us
that they relate to the standard normal probability
distribution.)
© 2003 South-Western/Thomson Learning™
Slide 15
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)
© 2003 South-Western/Thomson Learning™
=NORMSDIST(1)
=NORMSDIST(1)-NORMSDIST(0)
=NORMSDIST(1.25)-NORMSDIST(0)
=NORMSDIST(1)-NORMSDIST(-1)
=1-NORMSDIST(1.58)
=NORMSDIST(-0.5)
Slide 16
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)
© 2003 South-Western/Thomson Learning™
0.8413
0.3413
0.3944
0.6827
0.0571
0.3085
Slide 17
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
© 2003 South-Western/Thomson Learning™
=NORMSINV(0.9)
=NORMSINV(0.975)
=NORMSINV(0.025)
Slide 18
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
© 2003 South-Western/Thomson Learning™
1.28
1.96
-1.96
Slide 19
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 20 gallons, a replenishment order is placed.
The store manager is concerned that sales are being
lost due to stockouts while waiting for an order. It has
been determined that leadtime demand 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).

© 2003 South-Western/Thomson Learning™
Slide 20
Example: Pep Zone

Standard Normal Probability Distribution
The Standard Normal table shows an area of .2967 for
the region between the z = 0 and z = .83 lines below.
The shaded tail area is .5 - .2967 = .2033. The
probability of a stockout is .2033.
Area = .2967
z = (x - )/s
= (20 - 15)/6
= .83
Area = .5 - .2967
= .2033
Area = .5
0
© 2003 South-Western/Thomson Learning™
.83
z
Slide 21
Example: Pep Zone

z
Using the Standard Normal Probability Table
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753
.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
.4
.1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
.5
.1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
.6
.2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549
.7
.2580 .2612 .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
© 2003 South-Western/Thomson Learning™
Slide 22
Example: Pep Zone

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?
Area = .05
Area = .5 Area = .45
z.05
0
Let z.05 represent the z value cutting the .05 tail area.
© 2003 South-Western/Thomson Learning™
Slide 23
Example: Pep Zone

Using the Standard Normal Probability Table
We now look-up the .4500 area in the Standard
Normal Probability table to find the corresponding
z.05 value.
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 .4732 .4738 .4744 .4750 .4756 .4761 .4767
.
.
.
.
.
.
.
.
.
.
.
z.05 = 1.645 is a reasonable estimate.
© 2003 South-Western/Thomson Learning™
Slide 24
Example: Pep Zone

Standard Normal Probability Distribution
The corresponding value of x is given by
x =  + z.05s
= 15 + 1.645(6)
= 24.87
A reorder point of 24.87 gallons will place the
probability of a stockout during leadtime at .05.
Perhaps Pep Zone should set the reorder point at
25 gallons to keep the probability under .05.
© 2003 South-Western/Thomson Learning™
Slide 25
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.
© 2003 South-Western/Thomson Learning™
Slide 26
Using Excel to Compute Normal Probabilities

Formula Worksheet for Pep Zone Example
1
2
3
4
5
6
7
8
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)
© 2003 South-Western/Thomson Learning™
Slide 27
Using Excel to Compute Normal Probabilities

Value Worksheet for Pep Zone Example
1
2
3
4
5
6
7
8
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.
© 2003 South-Western/Thomson Learning™
Slide 28
Exponential Probability Distribution

Exponential Probability Density Function
f ( x) 
1

e  x /  for x > 0,  > 0
where:
 = mean
e = 2.71828
© 2003 South-Western/Thomson Learning™
Slide 29
Exponential Probability Distribution

Cumulative Exponential Distribution Function
P ( x  x0 )  1  e  xo / 
where:
x0 = some specific value of x
© 2003 South-Western/Thomson Learning™
Slide 30
Using Excel to Compute Exponential Probabilities


Excel’s EXPONDIST function can be used to compute
exponential probabilities.
The function has three arguments:
• First – the value of the random variable x
• Second – 1/m (the inverse of the mean number of
occurrences in an interval)
• Third – “TRUE” or “FALSE” (we will always enter
“TRUE” because we’re seeking a cumulative
probability)
© 2003 South-Western/Thomson Learning™
Slide 31
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)
© 2003 South-Western/Thomson Learning™
Slide 32
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
© 2003 South-Western/Thomson Learning™
Slide 33
Example: Al’s Carwash

Exponential Probability Distribution
The time between arrivals of cars at Al’s
Carwash 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.
P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866
© 2003 South-Western/Thomson Learning™
Slide 34
Example: Al’s Carwash

Graph of the Probability Density Function
f(x)
.4
.3
P(x < 2) = area = .4866
.2
.1
x
1
2
3
4
5
6
7
8
9 10
Time Between Successive Arrivals (mins.)
© 2003 South-Western/Thomson Learning™
Slide 35
Using Excel to Compute Exponential Probabilities

Formula Worksheet for Al’s Carwash Example
1
2
3
4
A
B
Probabilities: Exponential Distribution
P (x < 2)
=EXPONDIST(2,1/3,TRUE)
© 2003 South-Western/Thomson Learning™
Slide 36
Using Excel to Compute Exponential Probabilities

Value Worksheet for Al’s Carwash Example
1
2
3
4
A
B
Probabilities: Exponential Distribution
P (x < 2)
0.4866
© 2003 South-Western/Thomson Learning™
Slide 37
Relationship between the Poisson
and Exponential Distributions
(If) the Poisson distribution
provides an appropriate description
of the number of occurrences
per interval
(If) the exponential distribution
provides an appropriate description
of the length of the interval
between occurrences
© 2003 South-Western/Thomson Learning™
Slide 38
End of Chapter 6
© 2003 South-Western/Thomson Learning™
Slide 39