OPRE 6301-SYSM 6303 Chapter 08 -Students

8/17/2019 OPRE 6301-SYSM 6303 Chapter 08 -Students

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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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Relative Frequency divided by Class Width

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Determine the area of the histogram bars of therange of interest and add them together.

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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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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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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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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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What is the probability that daily sales will fallbetween 2,500 and 3,000 gallons?

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What is the probability that the service station will

sell at least 4,000 gallons?

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What is the probability that the service station willsell exactly 2,500 gallons?

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Normal ProbabilityDistribution

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Same variance with different means

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Same means with different variances

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Calculating normal probabilities

Use the standard normal random variable, Z:

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X Z


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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Let’s investigate

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?)100,1( X P


Let’s investigate

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?)0.1( Z P


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The Standard Normal Distribution Table

Z-table

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Let’s investigate

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8413.0)0.1( Z P

8413.0)100,1( X P


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Finding a Z-value for a given probability value

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8-37Z0.025

=1.96


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Z0.05

=1.645


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OPRE 6301-SYSM 6303 Chapter 08 -Students