Chapter 6 Continuous Probability Distributions

1
Chapter 6 Continuous Probability Distributions

• Uniform Probability Distribution
• Normal Probability Distribution
• Exponential Probability Distribution

f(x)
x
?
2
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.
• 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.

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Uniform Probability Distribution

• A random variable is uniformly distributed
  whenever the probability is proportional to the
  intervals length.
• Uniform Probability Density Function
• f(x) 1/(b - a) for a
• 0 elsewhere
• where a smallest value the variable can
  assume
• b largest value the variable can assume

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Uniform Probability Distribution

• Expected Value of x
• E(x) (a b)/2
• Variance of x
• Var(x) (b - a)2/12
• where a smallest value the variable can
  assume
• b largest value the variable can
  assume

5
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.
• The probability density function is
• f(x) 1/10 for 5
• 0 elsewhere
• where
• x salad plate filling weight

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Example Slater's Buffet

• Uniform Probability Distribution
• What is the probability that a customer will
  take between 12 and 15 ounces of salad?

f(x)
P(12
1/10
x
5
10
12
15
Salad Weight (oz.)
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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

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Normal Probability Distribution

• Graph of the Normal Probability Density Function

f(x)
x
?
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Normal Probability Distribution

• Characteristics of the Normal Probability
  Distribution
• The shape of the normal curve is often
  illustrated as a bell-shaped curve.
• Two parameters, m (mean) and s (standard
  deviation), determine the location and shape of
  the distribution.
• The highest point on the normal curve is at the
  mean, which is also the median and mode.
• The mean can be any numerical value negative,
  zero, or positive.
• continued

10
Normal Probability Distribution

• Characteristics of the Normal Probability
  Distribution
• The normal curve is symmetric.
• The standard deviation determines the width of
  the curve larger values result in wider, flatter
  curves.
• The total area under the curve is 1 (.5 to the
  left of the mean and .5 to the right).
• Probabilities for the normal random variable are
  given by areas under the curve.

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Normal Probability Distribution

• Percent () of Values in Some Commonly Used
  Intervals
• 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.

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Normal Probability Distribution

• Normal Probability Density Function
• where
• ? mean
• ? standard deviation
• ? 3.14159
• e 2.71828

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Standard Normal Probability Distribution

• A random variable that has a normal distribution
  with a mean of zero and a standard deviation of
  one is said to have a standard normal probability
  distribution.
• The letter z is commonly used to designate this
  normal random variable.
• Converting to the Standard Normal Distribution
• We can think of z as a measure of the number of
  standard deviations x is from ?.

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Using Excel to Compute Standard Normal
Probabilities

• Excel has two functions for computing
  probabilities and z values for a standard normal
  distribution
• NORMSDIST is used to compute the cumulative
  probability given a z value.
• NORMSINV is used to compute the z value given a
  cumulative probability.
• (The letter S in the above function names reminds
  us
• that they relate to the standard normal
  probability
• distribution.)

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Using Excel to ComputeStandard Normal
Probabilities

• Formula Worksheet

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Using Excel to ComputeStandard Normal
Probabilities

• Value Worksheet

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Using Excel to ComputeStandard Normal
Probabilities

• Formula Worksheet

18
Using Excel to ComputeStandard Normal
Probabilities

• Value Worksheet

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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 20 gallons, a replenishment order is
  placed.
• The store manager is concerned that sales are
  being
• lost due to stockouts while waiting for an order.
  It has
• been determined that leadtime demand 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).

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Example Pep Zone

• Standard Normal Probability Distribution
• The Standard Normal table shows an area of .2967
  for the region between the z 0 and z .83
  lines below. The shaded tail area is .5 - .2967
  .2033. The probability of a stock- out is
  .2033.
• z (x - ?)/?
• (20 - 15)/6
• .83

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Example Pep Zone

• Using the Standard Normal Probability Table

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Example Pep Zone

• 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?
• Let z.05 represent the z value cutting the .05
  tail area.

Area .05
Area .5
Area .45
z.05
0
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Example Pep Zone

• Using the Standard Normal Probability Table
• We now look-up the .4500 area in the Standard
  Normal Probability table to find the
  corresponding z.05 value.
• z.05 1.645 is a reasonable estimate.

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Example Pep Zone

• Standard Normal Probability Distribution
• The corresponding value of x is given by
• x ? z.05?
• ?? 15 1.645(6)
• 24.87
• A reorder point of 24.87 gallons will place the
  probability of a stockout during leadtime at .05.
  Perhaps Pep Zone should set the reorder point
  at 25 gallons to keep the probability under .05.

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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.

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Using Excel to Compute Normal Probabilities

• Formula Worksheet for Pep Zone Example

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Using Excel to Compute Normal Probabilities

• Value Worksheet for Pep Zone Example

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

• Exponential Probability Density Function
• for x 0, ? 0
• where ? mean
• e 2.71828

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Exponential Probability Distribution

• Cumulative Exponential Distribution Function
• where
• x0 some specific value of x

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Using Excel to Compute Exponential Probabilities

• Excels EXPONDIST function can be used to compute
  exponential probabilities.
• The function has three arguments
• First the value of the random variable x
• Second 1/m (the inverse of the mean number of
  occurrences in an interval)
• Third TRUE or FALSE (we will always enter
  TRUE because were seeking a cumulative
  probability)

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Using Excel to Compute Exponential Probabilities

• Formula Worksheet

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Using Excel to Compute Exponential Probabilities

• Value Worksheet

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Example Als Carwash

• Exponential Probability Distribution
• The time between arrivals of cars at Als
  Carwash 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.
• P(x

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Example Als Carwash

• Graph of the Probability Density Function

f(x)
.4
P(x
.3
.2
.1
x
1 2 3 4 5 6 7 8 9 10
Time Between Successive Arrivals (mins.)
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Using Excel to Compute Exponential Probabilities

• Formula Worksheet for Als Carwash Example

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Using Excel to Compute Exponential Probabilities

• Value Worksheet for Als Carwash Example

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Relationship between the Poissonand Exponential
Distributions
(If) the Poisson distribution provides an
appropriate description of the number of
occurrences per interval
(If) the exponential distribution provides an
appropriate description of the length of the
interval between occurrences