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Normal and Exponential Probability Distributions

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Normal and Exponential Probability Distributions Prof. Tom Willemain * * T.R. Willemain Analysis of dataset on MAU web site. Consider American Airlines flight 001 ... – PowerPoint PPT presentation

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Title: Normal and Exponential Probability Distributions


1
Normal and Exponential Probability Distributions
  • Prof. Tom Willemain

2
Normal Distribution
  • Also called Gaussian distribution

3
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4
Normal density curves
5
The standard normal distribution
6
Nonstandard normal distribution
7
Percentiles of normal distributions
8
Normal Probabilities and Percentiles in Minitab
9
In-Class Exercise
  • X N(15, 22).
  • Calculate the following
  • P(X lt 16) P(X 16) ?
  • 78th percentile of X ?

10
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11
Special Properties of Normal pdf
  • If X1 and X2 are normal RVs, then X1X2 also has
    a normal distribution.
  • If X1 N(µ1, s12) and X2 N(µ2, s22) AND if
    X1 and X2 are independent, then
  • X1X2 N(µ1µ2, s12s22)
  • i.e., means always add, and variances add if the
    components are independent

12
Normal probability plot
This is an informal, graphical test. If the data
plot close to a straight line, we are safe in
assuming normality.
13
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14
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15
Random Normal Data in Minitab
16
plot confirms that Minitab generated Normal data
17
Exponential Distribution
18
The exponential distribution
19
Exponential density curves
20
Memoryless Property of Exponential
  • Ex P X gt 100 X gt 90 PX gt 10, so its as
    if time restarted at 90.
  • This property is unique to the exponential and
    makes certain analyses easy.
  • We know that PX gt x exp(-?x) for all x 0
  • So what is PX gt x X gt c (for any x gt c)?
  • PX gt x AND X gt c/PX gt c
  • PX gt x/PX gt c
  • exp(-?x)/ exp(-?c) exp(-?x-c) QED

21
Exponential Distribution in Minitab
22
Exponential Probability Plot
23
In-Class Exercise
  • The time before failure of an electronic black
    box has an exponential distribution with mean
    150 hours.
  • Compute
  • Plifetime lt 500 hours
  • Plifetime is between 100 and 200 hours

24
In-Class Exercise
P100ltlifelt200 0.74-0.49 0.25
25
Demonstration of the Use ofRandom Numbers
toAnswer a More Complex Question
  • A black box has a mean lifetime of 150 hours.
    When it fails, it is replaced by another,
    identical unit.
  • There is a supply of 5 spare units on hand.
  • Estimate the pdf of the time until all the units
    are dead.

26
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