opre 6301-sysm 6303 chapter 08 -students
TRANSCRIPT
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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and
Uncertainty in Business
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Chapter EightContinuous Probability Distributions
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Probability Density Functions
What if we want to determine the probability of abill chosen at random being between 50 and 80?
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Probability Density Functions
Relative Frequency divided by Class Width
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Probability Density Functions
Determine the area of the histogram bars of therange of interest and add them together.
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Probability Density Functions
As the number of classes increases to a large
number, we can smooth the edges of therectangles to produce a smooth curve.
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Probability Density Functions
RequirementsApply to a probability density
function f(x) whose range is
The total area under the curvebetween a and b is 1.
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b xa
ba x x f and betweenallfor0)(
Uniform ProbabilityDistribution
Density Function
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b xawhereab
x f
1
)(
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Uniform ProbabilityDistribution
We can use the density function, f(x),to determine the area under the curve
between two values x1 and x 2The area under the curve is the probability
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Uniform ProbabilityDistribution
Let’s investigate
The amount of gasoline sold dailyat a service station is uniformly distributed
with a minimum of 2,000 gallons and
a maximum of 5,000 gallons
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Uniform ProbabilityDistribution
What is the probability that daily sales will fallbetween 2,500 and 3,000 gallons?
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Uniform ProbabilityDistribution
What is the probability that the service station will
sell at least 4,000 gallons?
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Uniform ProbabilityDistribution
What is the probability that the service station willsell exactly 2,500 gallons?
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Normal ProbabilityDistribution
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Normal ProbabilityDistribution
Same variance with different means
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Normal ProbabilityDistribution
Same means with different variances
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Normal ProbabilityDistribution
Calculating normal probabilities
Use the standard normal random variable, Z:
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X Z
Normal ProbabilityDistribution
Let’s investigate
Daily demand for gasoline at another gas stationis normally distributed with a mean of 1,000
gallons and a standard deviation of 100 gallons.
The owner has 1,100 gallons available to sell.What is the probability the owner will be able to
satisfy the day’s demand?
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Normal ProbabilityDistribution
Let’s investigate
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?)100,1( X P
Normal ProbabilityDistribution
Let’s investigate
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?)0.1( Z P
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Normal ProbabilityDistribution
The Standard Normal Distribution Table
Z-table
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Normal ProbabilityDistribution
Let’s investigate
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8413.0)0.1( Z P
8413.0)100,1( X P
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Normal ProbabilityDistribution
Finding a Z-value for a given probability value
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Normal ProbabilityDistribution
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=1.96
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Normal ProbabilityDistribution
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Z0.05
=1.645
Normal ProbabilityDistribution
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