1 1 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Chapter-6 Continuous Probability Distributions
Slide 2
2 2 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Learning Objectives n Identify Types of Continuous
Probability Distribution and Learn their respective Functional
Representation (Formulas) n Learn how to calculate the Mean and
Variance of a random Variable that assumes each o type of
Continuous Probability Distribution Identified.
Slide 3
3 3 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Continuous Probability Distributions n Uniform Probability
Distribution n Normal Probability Distribution n Exponential
Probability Distribution f ( x ) x x Uniform x Normal x x
Exponential
Slide 4
4 4 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Continuous Probability Distributions n A continuous random
variable can assume any value in an interval on the real line or in
a collection of intervals. n It is not possible to talk about the
probability of the random variable assuming a particular value. n
Instead, we talk about the probability of the random variable
assuming a value within a given interval.
Slide 5
5 5 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Continuous Probability Distributions n The probability of
the random variable assuming a value within some given interval
from x 1 to x 2 is defined to be the area under the graph of the
probability density function between x 1 and x 2. f ( x ) x x
Uniform x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 x Normal x1
x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 x1 x1x1 x1 x1 x1x1 x1 x2
x2x2 x2 x2 x2x2 x2 Exponential x x x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2
x2 x2x2 x2
Slide 6
6 6 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) 6.1) Uniform Continuous Probability Distribution where: a
= smallest value the variable can assume b = largest value the
variable can assume b = largest value the variable can assume f ( x
) = 1/( b a ) for a < x < b f ( x ) = 1/( b a ) for a < x
< b = 0 elsewhere = 0 elsewhere f ( x ) = 1/( b a ) for a < x
< b f ( x ) = 1/( b a ) for a < x < b = 0 elsewhere = 0
elsewhere n A random variable is uniformly distributed whenever the
probability is proportional to the intervals length. n The uniform
probability density function is:
Slide 7
7 7 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Var ( x ) = ( b - a ) 2 /12 E( x ) = ( a + b )/2 6.1)
Uniform Probability Distribution n Expected Value of x n Variance
of x
Slide 8
8 8 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) 6.1) Uniform Probability Distribution n Example: Slater's
Buffet 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.
Slide 9
9 9 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Uniform Probability Density Function f ( x ) = 1/10 for
5 < x < 15 f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere
= 0 elsewhere f ( x ) = 1/10 for 5 < x < 15 f ( x ) = 1/10
for 5 < x < 15 = 0 elsewhere = 0 elsewhere where: x = salad
plate filling weight x = salad plate filling weight 6.1) Uniform
Probability Distribution
Slide 10
10 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Expected Value of x n Variance of x E( x ) = ( a + b )/2
E( x ) = ( a + b )/2 = (5 + 15)/2 = (5 + 15)/2 = 10 = 10 E( x ) = (
a + b )/2 E( x ) = ( a + b )/2 = (5 + 15)/2 = (5 + 15)/2 = 10 = 10
Var( x ) = ( b - a ) 2 /12 Var( x ) = ( b - a ) 2 /12 = (15 5) 2
/12 = (15 5) 2 /12 = 8.33 = 8.33 Var( x ) = ( b - a ) 2 /12 Var( x
) = ( b - a ) 2 /12 = (15 5) 2 /12 = (15 5) 2 /12 = 8.33 = 8.33
Uniform Probability Distribution
Slide 11
11 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Uniform Probability Distribution for Salad Plate Filling
Weight f(x)f(x) f(x)f(x) x x 5 5 10 15 1/10 Salad Weight (oz.) 6.1)
Uniform Probability Distribution
Slide 12
12 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) f(x)f(x) f(x)f(x) x x 5 5 10 15 1/10 Salad Weight (oz.)
P(12 < x < 15) = 1/10(3) =.3 What is the probability that a
customer What is the probability that a customer will take between
12 and 15 ounces of salad? will take between 12 and 15 ounces of
salad? 12 6.1) Uniform Probability Distribution
Slide 13
13 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Probability Distribution
Slide 14
14 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) The normal probability distribution is the most important
distribution for describing a continuous random variable. It is
widely used in statistical inference. Normal Probability
Distribution
Slide 15
15 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Heights of people Heights Normal Probability Distribution
n It has been used in a wide variety of applications: Scientific
measurements measurementsScientific
Slide 16
16 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Amounts of rainfall Amounts Normal Probability
Distribution n It has been used in a wide variety of applications:
Test scores scoresTest
Slide 17
17 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Probability Distribution n Normal Probability
Density Function = mean = standard deviation = 3.14159 e = 2.71828
where:
Slide 18
18 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) The distribution is symmetric; its skewness The
distribution is symmetric; its skewness measure is zero. measure is
zero. The distribution is symmetric; its skewness The distribution
is symmetric; its skewness measure is zero. measure is zero. Normal
Probability Distribution n Characteristics x
Slide 19
19 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) The entire family of normal probability The entire family
of normal probability distributions is defined by its mean and its
distributions is defined by its mean and its standard deviation .
standard deviation . The entire family of normal probability The
entire family of normal probability distributions is defined by its
mean and its distributions is defined by its mean and its standard
deviation . standard deviation . Normal Probability Distribution n
Characteristics Standard Deviation Mean x
Slide 20
20 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) The highest point on the normal curve is at the The
highest point on the normal curve is at the mean, which is also the
median and mode. mean, which is also the median and mode. The
highest point on the normal curve is at the The highest point on
the normal curve is at the mean, which is also the median and mode.
mean, which is also the median and mode. Normal Probability
Distribution n Characteristics x
Slide 21
21 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Probability Distribution n Characteristics -10020
The mean can be any numerical value: negative, The mean can be any
numerical value: negative, zero, or positive. zero, or positive.
The mean can be any numerical value: negative, The mean can be any
numerical value: negative, zero, or positive. zero, or positive.
x
Slide 22
22 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Probability Distribution n Characteristics = 15 =
25 The standard deviation determines the width of the curve: larger
values result in wider, flatter curves. The standard deviation
determines the width of the curve: larger values result in wider,
flatter curves. x
Slide 23
23 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Probabilities for the normal random variable are
Probabilities for the normal random variable are given by areas
under the curve. The total area given by areas under the curve. The
total area under the curve is 1 (.5 to the left of the mean and
under the curve is 1 (.5 to the left of the mean and.5 to the
right)..5 to the right). Probabilities for the normal random
variable are Probabilities for the normal random variable are given
by areas under the curve. The total area given by areas under the
curve. The total area under the curve is 1 (.5 to the left of the
mean and under the curve is 1 (.5 to the left of the mean and.5 to
the right)..5 to the right). Normal Probability Distribution n
Characteristics.5.5 x
Slide 24
24 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Probability Distribution n Characteristics of
values of a normal random variable of values of a normal random
variable are within of its mean. are within of its mean. of values
of a normal random variable of values of a normal random variable
are within of its mean. are within of its mean.68.26%68.26% +/- 1
standard deviation of values of a normal random variable of values
of a normal random variable are within of its mean. are within of
its mean. of values of a normal random variable of values of a
normal random variable are within of its mean. are within of its
mean. 95.44%95.44% +/- 2 standard deviations of values of a normal
random variable of values of a normal random variable are within of
its mean. are within of its mean. of values of a normal random
variable of values of a normal random variable are within of its
mean. are within of its mean.99.72%99.72% +/- 3 standard
deviations
Slide 25
25 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Probability Distribution n Characteristics x 3 1 2
+ 1 + 2 + 3 68.26% 95.44% 99.72%
Slide 26
26 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) STANDARD NORMAL PROBABILITY DISTRIBUTION STANDARD NORMAL
PROBABILITY DISTRIBUTION A random variable that has a normal
distribution A random variable that has a normal distribution and a
mean of 0 and a standard deviation of 1 is and a mean of 0 and a
standard deviation of 1 is called a standard normal probability
distribution. called a standard normal probability distribution. A
random variable that has a normal distribution A random variable
that has a normal distribution and a mean of 0 and a standard
deviation of 1 is and a mean of 0 and a standard deviation of 1 is
called a standard normal probability distribution. called a
standard normal probability distribution. A Special Type of Normal
Distribution
Slide 27
27 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) 0 z The letter z is used to designate the standard The
letter z is used to designate the standard normal random variable.
normal random variable. The letter z is used to designate the
standard The letter z is used to designate the standard normal
random variable. normal random variable. Standard Normal
Probability Distribution
Slide 28
28 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Converting to the Standard Normal Distribution Standard
Normal Probability Distribution We can think of z as a measure of
the number of standard deviations x is from .
Slide 29
29 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Standard Normal Probability Distribution n Standard Normal
Density Function z = ( x )/ = 3.14159 e = 2.71828 where:
Slide 30
30 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Examples: Hands-on-Practice Problems
Slide 31
31 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Standard Normal Probability Distribution n Example: Pep
Zone 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. Pep Zone 5w-20 Motor
Oil
Slide 32
32 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) The store manager is concerned that sales are being 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 lead-time 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). Standard Normal Probability Distribution Pep Zone 5w-20
Motor Oil n Example: Pep Zone
Slide 33
33 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) z = ( x - )/ z = ( x - )/ = (20 - 15)/6 = (20 - 15)/6 =.83
=.83 z = ( x - )/ z = ( x - )/ = (20 - 15)/6 = (20 - 15)/6 =.83
=.83 n Solving for the Stockout Probability Step 1: Convert x to
the standard normal distribution. Pep Zone 5w-20 Motor Oil Step 2:
Find the area under the standard normal curve to the left of z
=.83. curve to the left of z =.83. Step 2: Find the area under the
standard normal curve to the left of z =.83. curve to the left of z
=.83. see next slide see next slide Standard Normal Probability
Distribution
Slide 34
34 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Cumulative Probability Table for the Standard Normal
Distribution Pep Zone 5w-20 Motor Oil P ( z
35 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) P ( z >.83) = 1 P ( z.83) = 1 P ( z .83) = 1 P ( z.83)
= 1 P ( z 20) Standard Normal Probability Distribution
Slide 36
36 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Solving for the Stockout Probability 0.83 Area =.7967
Area = 1 -.7967 =.2033 =.2033 z Pep Zone 5w-20 Motor Oil Standard
Normal Probability Distribution
Slide 37
37 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n 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? Pep Zone 5w-20 Motor Oil
Standard Normal Probability Distribution
Slide 38
38 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil 0
Area =.9500 Area =.0500 z z.05 Standard Normal Probability
Distribution
Slide 39
39 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil
Step 1: Find the z -value that cuts off an area of.05 in the right
tail of the standard normal in the right tail of the standard
normal distribution. distribution. Step 1: Find the z -value that
cuts off an area of.05 in the right tail of the standard normal in
the right tail of the standard normal distribution. distribution.
We look up the complement of the tail area (1 -.05 =.95) Standard
Normal Probability Distribution
Slide 40
40 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil
Step 2: Convert z.05 to the corresponding value of x. x = + z.05 x
= + z.05 = 15 + 1.645(6) = 24.87 or 25 = 24.87 or 25 x = + z.05 x =
+ z.05 = 15 + 1.645(6) = 24.87 or 25 = 24.87 or 25 A reorder point
of 25 gallons will place the probability A reorder point of 25
gallons will place the probability of a stockout during leadtime at
(slightly less than).05. of a stockout during leadtime at (slightly
less than).05. Standard Normal Probability Distribution
Slide 41
41 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil
By raising the reorder point from 20 gallons to 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 This is a significant
decrease in the chance that Pep Zone will be out of stock and
unable to meet a customers desire to make a purchase. Standard
Normal Probability Distribution
Slide 42
42 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Approximation of Binomial Probabilities
Slide 43
43 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Approximation of Binomial Probabilities When the
number of trials, n, becomes large, When the number of trials, n,
becomes large, we can use the normal distribution to approximate
the binomial probability distribution That is, the normal
probability distribution also serves as a means to approximation of
binomial probabilities: That is, the normal probability
distribution also serves as a means to approximation of binomial
probabilities: Pre-conditions : Pre-conditions : n > 20, np >
5, and n (1 - p ) > 5. n > 20, np > 5, and n (1 - p ) >
5.
Slide 44
44 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Normal Approximation of Binomial Probabilities Set = np
Set = np n When using the normal distribution to approximate
binomial probabilities, however, we add and a subtract 0.5 as a
correction factor. n That is, 0.5 is used as a continuity
correction factor because a continuous distribution is being used
to approximate a discrete distribution. n For example, P ( x = 10)
is approximated by P (9.5 < x < 10.5).
Slide 45
45 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Exponential Probability Distribution
Slide 46
46 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Exponential Probability Distribution n Often times, the
exponential probability distribution describes the time it takes to
complete a task. n The exponential random variables can be used to
describe: Time between vehicle arrivals at a toll booth Time
between vehicle arrivals at a toll booth Time required to complete
a questionnaire Time required to complete a questionnaire Distance
between major defects in a highway Distance between major defects
in a highway SLOW
Slide 47
47 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Density Function Exponential Probability Distribution
where: = mean e = 2.71828 e = 2.71828 for x > 0, > 0
Slide 48
48 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Exponential Probability Distribution where: x 0 = some
specific value of x x 0 = some specific value of x Cumulative
Probabilities Cumulative Probabilities
Slide 49
49 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Exponential Probability Distribution The exponential
distribution is skewed to the right. The exponential distribution
is skewed to the right. The skewness measure for the exponential
distribution The skewness measure for the exponential distribution
is 2. is 2. The skewness measure for the exponential distribution
The skewness measure for the exponential distribution is 2. is 2.
x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 Exponential x x f ( x )
x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2
Slide 50
50 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Relationship between the Poisson and Exponential
Distributions The Poisson distribution provides an appropriate
description of the number of occurrences per interval 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 The exponential distribution provides an appropriate
description of the length of the interval between occurrences x1
x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 Exponential x x f ( x ) x1
x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2
Slide 51
51 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) n Hands-on-Practice Problem
Slide 52
52 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Exponential Probability Distribution n Example: Als
Full-Service Pump The time between arrivals of cars The time
between arrivals of cars at Als 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.
Slide 53
53 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) x x f(x)f(x) f(x)f(x).1.3.4.2 1 2 3 4 5 6 7 8 9 10 Time
Between Successive Arrivals (mins.) Exponential Probability
Distribution P ( x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866 P (
x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866
Slide 54
54 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr.
Tadesse) Exponential Probability Distribution A property of the
exponential distribution is that A property of the exponential
distribution is that the mean, , and standard deviation, , are
equal. the mean, , and standard deviation, , are equal. A property
of the exponential distribution is that A property of the
exponential distribution is that the mean, , and standard
deviation, , are equal. the mean, , and standard deviation, , are
equal. Thus, the standard deviation, , and variance, 2, for the
time between arrivals at Als full-service pump are: = = 3 minutes 2
= (3) 2 = 9
