Transcript CHAPTER 6 CONTINUOUS PROBABILITY DISTRIBUTIONS

Continuous Probability Distributions



f (x)
Uniform Probability Distribution
Normal Probability Distribution
Exponential Probability Distribution
f (x) Exponential
Uniform
f (x)
Normal
x
x
x
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.
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
x
x
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
Uniform Probability Distribution

Expected Value of x
E(x) = (a + b)/2

Variance of x
Var(x) = (b - a)2/12
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.
Example: Slater's Buffet

Uniform Probability Density Function
f(x) = 1/10 for 5 < x < 15
=0
elsewhere
where:
x = salad plate filling weight
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
Example: Slater's Buffet

Uniform Probability Distribution
for Salad Plate Filling Weight
f(x)
1/10
5
10
15
Salad Weight (oz.)
x
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.)
x
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.
Normal Probability Distribution

It has been used in a wide variety of applications:
Heights
of people
Scientific
measurements
Normal Probability Distribution

It has been used in a wide variety of applications:
Test
scores
Amounts
of rainfall
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
Normal Probability Distribution

Characteristics
The distribution is symmetric, and is bell-shaped.
x
Normal Probability Distribution

Characteristics
The entire family of normal probability
distributions is defined by its mean  and its
standard deviation  .
Standard Deviation 
Mean 
x
Normal Probability Distribution

Characteristics
The highest point on the normal curve is at the
mean, which is also the median and mode.
x
Normal Probability Distribution

Characteristics
The mean can be any numerical value: negative,
zero, or positive.
x
-10
0
20
Normal Probability Distribution

Characteristics
The standard deviation determines the width of the
curve: larger values result in wider, flatter curves.
 = 15
 = 25
x
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
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.
Normal Probability Distribution

Characteristics
99.72%
95.44%
68.26%
 – 3
 – 1
 – 2

 + 3
 + 1
 + 2
x
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.
Standard Normal Probability Distribution
The letter z is used to designate the standard
normal random variable.
1
z
0
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 .
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.)
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)
=NORMSDIST(1)
=NORMSDIST(1)-NORMSDIST(0)
=NORMSDIST(1.25)-NORMSDIST(0)
=NORMSDIST(1)-NORMSDIST(-1)
=1-NORMSDIST(1.58)
=NORMSDIST(-0.5)
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)
0.8413
0.3413
0.3944
0.6827
0.0571
0.3085
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
=NORMSINV(0.9)
=NORMSINV(0.975)
=NORMSINV(0.025)
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
1.28
1.96
-1.96
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
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).
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
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)
.
.
.
.
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
P(x > 20)
Example: Pep Zone
Pep
Zone

5w-20
Motor Oil
Solving for the Stockout Probability
Area = .5 - .2967
Area = .2967
= .2033
0
.83
z
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?
Example: Pep Zone
Pep
Zone
5w-20
Motor Oil

Solving for the Reorder Point
Area = .4500
Area = .0500
0
z.05
z
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)
.
.
.
.
.
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.
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.
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.
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)
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.
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
Distance between
major defects
in a highway
Exponential Probability Distribution

Density Function
f ( x) 
1

e  x /  for x > 0,  > 0
where:
 = mean
e = 2.71828
Exponential Probability Distribution

Cumulative Probabilities
P ( x  x0 )  1  e  xo / 
where:
x0 = some specific value of x
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
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)
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
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.
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.)
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)
Using Excel to Compute
Exponential Probabilities

Value Worksheet
1
2
3
4
A
B
Probabilities: Exponential Distribution
P (x < 2)
0.4866
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