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Slides by
JohnLoucks
St. Edward’sUniversity
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Chapter 6 Continuous Probability Distributions
Uniform Probability Distribution
f (x)
x
Uniform
x
f (x) Normal
x
f (x) Exponential
Normal Probability Distribution Exponential Probability Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
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Continuous Probability Distributions
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)
x
Uniform
x1 x2
x
f (x) Normal
x1 x2
x1 x2
Exponential
x
f (x)
x1
x2
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Uniform Probability Distribution
where: a = smallest value the variable can assume b = largest value the variable can assume
f (x) = 1/(b – a) for a < x < b = 0 elsewhere
A random variable is uniformly distributed whenever the probability is proportional to the interval’s length.
The uniform probability density function is:
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Var(x) = (b - a)2/12
E(x) = (a + b)/2
Uniform Probability Distribution
Expected Value of x
Variance of x
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Uniform Probability Distribution
Example: Slater's Buffet Slater customers are charged for the
amount ofsalad they take. Sampling suggests that the
amountof salad taken is uniformly distributed
between 5ounces and 15 ounces.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Uniform Probability Density Function
f(x) = 1/10 for 5 < x < 15 = 0 elsewhere
where: x = salad plate filling weight
Uniform Probability Distribution
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Expected Value of x
E(x) = (a + b)/2 = (5 + 15)/2 = 10
Var(x) = (b - a)2/12 = (15 – 5)2/12 = 8.33
Uniform Probability Distribution
Variance of x
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Uniform Probability Distributionfor Salad Plate Filling Weight
f(x)
x
1/10
Salad Weight (oz.)
Uniform Probability Distribution
5 10 150
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or duplicated, or posted to a publicly accessible website, in whole or in part.
f(x)
x
1/10
Salad Weight (oz.)5 10 150
P(12 < x < 15) = 1/10(3) = .3
What is the probability that a customer will take between 12 and 15 ounces of salad?
Uniform Probability Distribution
12
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Area as a Measure of Probability
The area under the graph of f(x) and probability are identical.
This is valid for all continuous random variables. The probability that x takes on a value between some lower value x1 and some higher value x2 can be found by computing the area under the graph of f(x) over the interval from x1 to x2.
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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. It has been used in a wide variety of
applications including:• Heights of
people• Rainfall
amounts
• Test scores• Scientific
measurements Abraham de Moivre, a French mathematician, published The Doctrine of Chances in 1733.
He derived the normal distribution.
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Normal Probability Distribution
Normal Probability Density Function
2 2( ) / 21( ) 2xf x e
= mean = standard deviation = 3.14159e = 2.71828
where:
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The distribution is symmetric; its skewness measure is zero.
Normal Probability Distribution
Characteristics
x
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The entire family of normal probability distributions is defined by its mean and its standard deviation .
Normal Probability Distribution
Characteristics
Standard Deviation
Mean x
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The highest point on the normal curve is at the mean, which is also the median and mode.
Normal Probability Distribution
Characteristics
x
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Normal Probability Distribution
Characteristics
-10 0 25
The mean can be any numerical value: negative, zero, or positive.
x
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Normal Probability Distribution
Characteristics
= 15
= 25
The standard deviation determines the width of thecurve: larger values result in wider, flatter curves.
x
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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).
Normal Probability Distribution
Characteristics
.5 .5x
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Normal Probability Distribution
Characteristics (basis for the empirical rule)
of values of a normal random variable are within of its mean.68.26%
+/- 1 standard deviation
of values of a normal random variable are within of its mean.95.44%
+/- 2 standard deviations
of values of a normal random variable are within of its mean.99.72%
+/- 3 standard deviations
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Normal Probability Distribution
Characteristics (basis for the empirical rule)
x – 3 – 1
– 2 + 1
+ 2 + 3
68.26%95.44%99.72%
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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.
Characteristics
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1
0z
The letter z is used to designate the standard normal random variable.
Standard Normal Probability Distribution
Characteristics
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Converting to the Standard Normal Distribution
Standard Normal Probability Distribution
z x
We can think of z as a measure of the number ofstandard deviations x is from .
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is used to compute the z value given a cumulative probability.NORMSINVNORM.S.INV
is used to compute the cumulative probability given a z value.NORMSDISTNORM.S.DIST
Using Excel to ComputeStandard Normal Probabilities
Excel has two functions for computing probabilities and z values for a standard normal distribution:
The “S” in the function names remindsus that they relate to the standardnormal probability distribution.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Formula Worksheet
Using Excel to ComputeStandard Normal Probabilities
A B12 3 P (z < 1.00) =NORM.S.DIST(1)4 P (0.00 < z < 1.00) =NORM.S.DIST(1)-NORM.S.DIST(0)5 P (0.00 < z < 1.25) =NORM.S.DIST(1.25)-NORM.S.DIST(0)6 P (-1.00 < z < 1.00) =NORM.S.DIST(1)-NORM.S.DIST(-1)7 P (z > 1.58) =1-NORM.S.DIST(1.58)8 P (z < -0.50) =NORM.S.DIST(-0.5)9
Probabilities: Standard Normal Distribution
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Excel Value Worksheet
Using Excel to ComputeStandard Normal Probabilities
A B12 3 P (z < 1.00) 0.84134 P (0.00 < z < 1.00) 0.34135 P (0.00 < z < 1.25) 0.39446 P (-1.00 < z < 1.00) 0.68277 P (z > 1.58) 0.05718 P (z < -0.50) 0.30859
Probabilities: Standard Normal Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Formula Worksheet
Using Excel to ComputeStandard Normal Probabilities
A B12 3 z value with .10 in upper tail =NORM.S.INV(0.9)4 z value with .025 in upper tail =NORM.S.INV(0.975)5 z value with .025 in lower tail =NORM.S.INV(0.025)6
Finding z Values, Given Probabilities
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Value Worksheet
Using Excel to ComputeStandard Normal Probabilities
A B12 3 z value with .10 in upper tail 1.284 z value with .025 in upper tail 1.965 z value with .025 in lower tail -1.966
Finding z Values, Given Probabilities
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Standard Normal Probability Distribution
Example: Pep Zone Pep Zone sells auto parts and supplies
includinga popular multi-grade motor oil. When the
stock ofthis oil drops to 20 gallons, a replenishment
order isplaced. The store manager is concerned that sales
arebeing lost due to stockouts while waiting for areplenishment order.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
Standard Normal Probability Distribution
Example: Pep Zone
The manager would like to know the probability
of a stockout during replenishment lead-time. In
other words, what is the probability that demand
during lead-time will exceed 20 gallons? P(x > 20) = ?
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or duplicated, or posted to a publicly accessible website, in whole or in part.
z = (x - )/ = (20 - 15)/6 = .83
Solving for the Stockout Probability
Step 1: Convert x to the standard normal distribution.
Step 2: Find the area under the standard normal curve to the left of z = .83.
see next slide
Standard Normal Probability Distribution
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z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09. . . . . . . . . . ..5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389. . . . . . . . . . .
Cumulative Probability Table for the Standard Normal Distribution
P(z < .83)
Standard Normal Probability Distribution
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P(z > .83) = 1 – P(z < .83) = 1- .7967 = .2033
Solving for the Stockout Probability
Step 3: Compute the area under the standard normal curve to the right of z = .83.
Probability of a
stockoutP(x > 20)
Standard Normal Probability Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Solving for the Stockout Probability
0 .83
Area = .7967Area = 1 - .7967 = .2033
z
Standard Normal Probability Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Standard Normal Probability Distribution
Standard Normal Probability Distribution
If the manager of Pep Zone wants the probability
of a stockout during replenishment lead-time to be
no more than .05, what should the reorder point be?
---------------------------------------------------------------
(Hint: Given a probability, we can use the standard
normal table in an inverse fashion to find thecorresponding z value.)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Solving for the Reorder Point
0
Area = .9500
Area = .0500
zz.05
Standard Normal Probability Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09. . . . . . . . . . .
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .94411.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .95451.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .96331.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .97061.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 . . . . . . . . . . .
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.
We look upthe
complement of the tail area(1 - .05 = .95)
Standard Normal Probability Distribution
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Solving for the Reorder Point
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.
Standard Normal Probability Distribution
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Normal Probability Distribution
Solving for the Reorder Point
15x
24.87
Probability of a
stockout during
replenishmentlead-time
= .05
Probability of no
stockout during
replenishmentlead-time
= .95
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Solving for the Reorder Point By raising the reorder point from 20 gallons to 25 gallons on hand, the probability of a stockoutdecreases from about .20 to .05. This is a significant decrease in the chance thatPep Zone will be out of stock and unable to meet acustomer’s desire to make a purchase.
Standard Normal Probability Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Using Excel to ComputeNormal Probabilities
Excel has two functions for computing cumulative probabilities and x values for any normal distribution:NORM.DIST is used to compute the cumulativeprobability given an x value.
NORM.INV is used to compute the x value givena cumulative probability.
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Excel Formula Worksheet
Using Excel to ComputeNormal Probabilities
A B12 3 P (x > 20) =1-NORM.DIST(20,15,6,TRUE)4 56 7 x value with .05 in upper tail =NORM.INV(0.95,15,6)8
Probabilities: Normal Distribution
Finding x Values, Given Probabilities
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Formula Worksheet
Using Excel to ComputeNormal Probabilities
A B12 3 P (x > 20) 0.20234 56 7 x value with .05 in upper tail 24.878
Probabilities: Normal Distribution
Finding x Values, Given Probabilities
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.
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Exponential Probability Distribution
The exponential probability distribution is useful in describing the time it takes to complete a task.
• Time between vehicle arrivals at a toll booth• Time required to complete a questionnaire• Distance between major defects in a highway
The exponential random variables can be used to describe:
In waiting line applications, the exponential distribution is often used for service times.
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Exponential Probability Distribution
A property of the exponential distribution is that the mean and standard deviation are equal. The exponential distribution is skewed to the right. Its skewness measure is 2.
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Density Function
Exponential Probability Distribution
where: = expected or mean e = 2.71828
f x e x( ) / 1
for x > 0
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Cumulative Probabilities
Exponential Probability Distribution
P x x e x( ) / 0 1 o
where: x0 = some specific value of x
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Using Excel to ComputeExponential Probabilities
The EXPON.DIST function can be used to compute exponential probabilities.
The EXPON.DIST function has three arguments: 1st The value of the random variable x
2nd 1/m
3rd “TRUE” or “FALSE” the inverse of the meannumber of occurrences in an intervalWe will always enter
“TRUE” because we’re seeking a cumulative
probability.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Using Excel to ComputeExponential Probabilities
Excel Formula WorksheetA B
12 3 P (x < 18) =EXPON.DIST(18,1/15,TRUE)4 P (6 < x < 18) =EXPON.DIST(18,1/15,TRUE)-EXPON.DIST(6,1/15,TRUE)5 P (x > 8) =1-EXPON.DIST(8,1/15,TRUE)6
Probabilities: Exponential Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Using Excel to ComputeExponential Probabilities
Excel Value WorksheetA B
12 3 P (x < 18) 0.69884 P (6 < x < 18) 0.36915 P (x > 8) 0.58666
Probabilities: Exponential Distribution
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Exponential Probability Distribution
Example: Al’s Full-Service Pump The time between arrivals of cars at Al’s
full-service gas pump follows an exponential
probabilitydistribution with a mean time between arrivals
of 3minutes. Al would like to know the probability
thatthe time between two successive arrivals will
be 2minutes or less.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
x
f(x)
.1
.3
.4
.2
0 1 2 3 4 5 6 7 8 9 10Time Between Successive Arrivals (mins.)
Exponential Probability Distribution
P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866
Example: Al’s Full-Service Pump
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Formula Worksheet
Using Excel to ComputeExponential Probabilities
A B12 3 P (x < 2) =EXPON.DIST(2,1/3,TRUE)4
Probabilities: Exponential Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Excel Value Worksheet
Using Excel to ComputeExponential Probabilities
A B12 3 P (x < 2) 0.48664
Probabilities: Exponential Distribution
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Relationship between the Poissonand Exponential Distributions
The Poisson distributionprovides an appropriate description
of the number of occurrencesper interval
The exponential distributionprovides an appropriate description
of the length of the intervalbetween occurrences
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End of Chapter 6