Continuous Probability Distributions

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Continuous Probability Distributions
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Continuous Probability Distributions

• The Uniform Distribution

• The Normal Distribution

• The Exponential Distribution

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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 continuous random variable
  assuming a specific value is 0.
• 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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Continuous Probability Distributions
?
b
a
x1
P(x1 x x2)
P(x x1)
P(x x1) 1- P(xP(x x1)
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The Uniform Probability Distribution

• 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
• The probability of the continuous random variable
  assuming a specific value is 0.
• P(xx1) 0

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Example Slater's Buffet

• 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.

• Probability Density Function

f (x ) 1/10 for 5 elsewhere where x salad plate filling
weight
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Example Slater's Buffet

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

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The Uniform Probability Distribution
f (x )
P(81/10
x
5
8
12
15
P(8 10
The Uniform Probability Distribution
f (x )
P(01/10
x
5
12
15
P(0 11
The Uniform Probability Distribution

• Uniform Probability Density Function
• f (x) 1/(b - a) for a
• 0 elsewhere
• 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

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Normal Distribution
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Before Starting Normal Distribution
P(8P( P( P(8 14
The Normal Probability Distribution

• Graph of the Normal Probability Density Function

f (x )
x
?
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The Normal Curve

• The shape of the normal curve is often
  illustrated as a bell-shaped curve.
• The highest point on the normal curve is at the
  mean of the distribution.
• The normal curve is symmetric.
• The standard deviation determines the width of
  the curve.

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The Normal Curve

• The total area under the curve the same as any
  other probability distribution is 1.
• The probability of the normal random variable
  assuming a specific value the same as any other
  continuous probability distribution is 0.
• Probabilities for the normal random variable are
  given by areas under the curve.

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The Normal Probability Density Function
where ? mean ? standard
deviation ? 3.14159 e 2.71828
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The Standard Normal Probability Density Function
where ? 0 ? 1 ?
3.14159 e 2.71828
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Given any positive value for z, the table will
give us the following probability
The probability that we find using the table is
the probability of having a standard normal
variable between 0 and the given positive z.
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Given z find the probability
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Given any probability between 0 and .5,, the
table will give us the following positive z value

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Given the probability find z find
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What is the z value where probability of a
standard normal variable to be greater than z is
.1
10
40
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Standard Normal Probability Distribution(Z
Distribution)
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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.
• The following expression convert any Normal
  Distribution into the Standard Normal
  Distribution

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Example Pep Zone

• Pep Zone sells auto parts and supplies including
  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.
• In Summary we have a N (15, 6) A normal random
• variable with mean of 15 and std of 6.
• The manager would like to know the probability of
  a stockout, P(x 20).

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Standard Normal Distribution

• z (x - ? )/?
• (20 - 15)/6
• .83

Area .5
z
0
.83
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Example Pep Zone
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The Probability of Demand Exceeding 20
The Standard Normal table shows an area of .2967
for the region between the z 0 line and the z
.83 line above. The shaded tail area is .5 -
.2967 .2033. The probability of a stockout is
.2033.
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Example Pep Zone

• 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 tail
  area of .05.

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

• 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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Example Aptitude Test
A firm has assumed that the distribution of the
aptitude test of people applying for a job in
this firm is normal. The following sample is
available.
71 66 61 65 54 93 60 86 70 70 73 73 55 63 56 62
76 54 82 79 76 68 53 58 85 80 56 61 61 64 65 6
2 90 69 76 79 77 54 64 74 65 65 61 56 63 80 56 7
1 79 84
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Example Mean and Standard Deviation
We first need to estimate mean and standard
deviation
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z Values
What test mark has the property of having 10 of
test marks being less than or equal to it To
answer this question, we should first answer the
following What is the standard normal value (z
value), such that 10 of z values are less than
or equal to it?
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z Values
We need to use standard Normal distribution in
Table 1.
10
10
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z Values
10
40
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z Values
40
z 1.28
10
z - 1.28
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z Values and x Values
The standard normal value (z value), such that
10 of z values are less than or equal to it is z
-1.28
To transform this standard normal value to a
similar value in our example, we use the
following relationship
The normal value of test marks such that 10 of
random variables are less than it is 55.1.
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z Values and x Values
Following the same procedure, we could find z
values for cases where 20, 30, 40, of random
variables are less than these values. Following
the same procedure, we could transform z values
into x values.
Lower 10 -1.28 55.1 Lower 20 -.84 59.68 Lowe
r 30 -.52 63.01 Lower 40 -.25 65.82 Lower
50 0 68.42 Lower 60 .25 71.02
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Example Victor Computers

• Victor Computers manufactures and sells a
• general purpose microcomputer. As part of a
  study to evaluate sales personnel, management
  wants to determine if the annual sales volume
  (number of units sold by a salesperson) follows a
  normal probability distribution.
• A simple random sample of 30 of the
  salespeople was taken and their numbers of units
  sold are below.
• 33 43 44 45 52 52 56 58
  63 64
• 64 65 66 68 70 72 73 73
  74 75
• 83 84 85 86 91 92 94 98
  102 105
• (mean 71, standard deviation 18.54)
• Partition this Normal distribution into 6 equal
  probability parts

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Example Victor Computers
Areas 1.00/6 .1667
71
53.02
88.98 71 .97(18.54)
63.03
78.97
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The End
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Normal Approximation of Binomial Probabilities

• When the number of trials, n, becomes large,
  evaluating the binomial probability function by
  hand or with a calculator is difficult.
• The normal probability distribution provides an
  easy-to-use approximation of binomial
  probabilities where n 20, np 5, and n(1 - p)
  5.
• Set ? np
• Add and subtract a continuity correction factor
  because a continuous distribution is being used
  to approximate a discrete distribution. For
  example,
• P(x 10) is approximated by P(9.5

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The Exponential Probability Distribution

• Exponential Probability Density Function
• for x 0, ? 0
• where ? mean
• e 2.71828
• Cumulative Exponential Distribution Function
• where x0 some specific value of x

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Example Als Carwash

• 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

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The End