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PPT – Normal and Exponential Probability Distributions PowerPoint presentation | free to download - id: 41d5f6-MmUwM

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Normal and Exponential Probability Distributions

- Prof. Tom Willemain

Normal Distribution

- Also called Gaussian distribution

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Normal density curves

The standard normal distribution

Nonstandard normal distribution

Percentiles of normal distributions

Normal Probabilities and Percentiles in Minitab

In-Class Exercise

- X N(15, 22).
- Calculate the following
- P(X lt 16) P(X 16) ?
- 78th percentile of X ?

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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

Normal probability plot

This is an informal, graphical test. If the data

plot close to a straight line, we are safe in

assuming normality.

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Random Normal Data in Minitab

plot confirms that Minitab generated Normal data

Exponential Distribution

The exponential distribution

Exponential density curves

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

Exponential Distribution in Minitab

Exponential Probability Plot

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

In-Class Exercise

P100ltlifelt200 0.74-0.49 0.25

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.

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