ETM 620 - 09U1

Continuous probability distributionsExamples:

The probability that the average daily temperature in Georgia during the month of August falls between 90 and 95 degrees is

__________

The probability that a given part will fail before 1000 hours of use is

__________

In general, __________

Understanding continuous distributions

The probability that the average daily temperature in Georgia during the month of August falls between 90 and 95 degrees is

The probability that a given part will fail before 1000 hours of use is

-5 -3 -1 1 3 5

0 5 10 15 20 25 30

ETM 620 - 09U2

Continuous probability distributionsThe continuous probability density function (pdf)

f(x) ≥ 0, for all x ∈ R

The cumulative distribution, F(x)

Expectation and variance

μ = E(X) = ______________σ2 = ________________

1)( dxxf

b

a

dxxfbXaP )()(

x

dttfxXPxF )()()(

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An example

x, 0 < x < 1f(x) = 2-x, 1 ≤ x < 2

0, elsewhere

1st – what does the function look like?

a) P(X < 1.2) = ___________________

b) P(0.5 < X < 1) = ___________________

c) E(X) = -∞∫∞ x f(x) dx = ________________________ETM 620 - 09U4

{

Continuous probability distributionsMany continuous probability distributions,

including:

Uniform

Exponential

Gamma

Weibull

Normal

Lognormal

Bivariate normalETM 620 - 09U5

Uniform distributionSimplest – characterized by the interval

endpoints, A and B.

A ≤ x ≤ B

= 0 elsewhere

Mean and variance:

and

ETM 620 - 09U6

ABBAxf

1),;(

2BA

12

)( 22 AB

ExampleA circuit board failure causes a shutdown of a computing system until a new board is delivered. The delivery time X is uniformly distributed between 1 and 5 days.

What is the probability that it will take 2 or more days for the circuit board to be delivered?

ETM 620 - 09U7

Gamma & Exponential distributionsRecall the Poisson Process

Number of occurrences in a given interval or region

“Memoryless” processSometimes we’re interested in the time or

area until a certain number of events occur.For example

An average of 2.7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process.

What is the probability that up to a minute will elapse before 2 calls arrive?

How long before the next call?

ETM 620 - 09U8

Gamma distributionThe density function of the random variable X

with gamma distribution having parameters α (number of occurrences) and β (time or region, 1/λ).

x > 0.Gamma Distribution

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

x

f(x)

ETM 620 - 09U9

22

)!1()(

)(

1)( 1

nn

exxfx

Exponential distribution

0 5 10 15 20 25 30

x

exf

1

)(

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Special case of the gamma distribution with α = 1.

x > 0.

Describes the time until or time between Poisson events.

μ = β

σ2 = β2

ExampleAn average of 2.7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process.

What is the probability that up to a minute will elapse before 2 calls arrive?

β = ________ α = ________ P(X ≤ 1) = _________________________________

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Example (cont.)What is the expected time before the next call arrives?

β = ________ α = ________ μ = _________________________________

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Another exampleExample 6-6, page 136.

Note: in Excel, use =GAMMADIST(x,alpha,beta,cumulative) to find the probability associated with a specific value of x. Use =GAMMAINV(probability,alpha,beta) to find the value of x given a probability. ETM 620 - 09U13

Wiebull Distribution

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Used for many of the same applications as the gamma and exponential distributions, but does not require memoryless property of the

exponential

x

x

exF

xexxf

1)(

0,),;( 1

ExampleDesigners of wind turbines for power generation are interested in accurately describing variations in wind speed, which in a certain location can be described using the Weibull distribution with α = 0.02 and β = 2. A designer is interested in determining the probability that the wind speed in that location is between 3 and 7 mph.

P(3 < X < 7) = ___________________________

In Excel, =WEIBULL(x,alpha,beta,cumulative)ETM 620 - 09U15

Normal distributionThe “bell-shaped curve”Also called the Gaussian distributionThe most widely used distribution in statistical

analysisforms the basis for most of the parametric tests

we’ll perform later in this course.describes or approximates most phenomena in

nature, industry, or researchRandom variables (X) following this distribution

are called normal random variables.the parameters of the normal distribution are μ

and σ (sometimes μ and σ2.)

ETM 620 - 09U16

Normal distributionThe density function of the normal random

variable X, with mean μ and variance σ2, is

, all x.

In Excel, =NORMDIST(x,mean,standard_dev,cumulative) and =NORMINV(probability,mean,standard_dev) ETM 620 - 09U17

2

2

2)(

2

1),;(

x

exn

Normal Distribution

00.05

0.10.150.2

0.25

0.30.35

0 2 4 6 8 10 12

x

P(x

)

(μ = 5, σ = 1.5)

Standard normal RV …Note: the probability of X taking on any

value between x1 and x2 is given by:

To ease calculations, we define a normal random variable

where Z is normally distributed with μ = 0 and σ2 = 1 and,

ETM 620 - 09U18

dxedxxnxXxPx

x

xx

x

2

1

2

22

1

2)(

21 2

1),;()(

X

Z

)(2

1)( 2 ZdxezZP

z x

Standard normal distribution

Standard Normal Distribution

-5 -4 -3 -2 -1 0 1 2 3 4 5

Z

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Table II, pg. 601-602: “Cumulative Standard Normal Distribution”

In Excel, =NORMSDIST(z) and =NORMSINV(probability)

ExamplesP(Z ≤ 1) = Φ(1) =

P(Z ≥ -1) =

P(-0.45 ≤ Z ≤ 0.36) =

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-5 0 5

-5 0 5

-5 0 5

Your turn …Use Excel to determine (draw the picture!)

1. P(Z ≤ 0.8) =

2. P(Z ≥ 1.96) =

3. P(-0.25 ≤ Z ≤ 0.15) =

4. P(Z ≤ -2.0 or Z ≥ 2.0) =

ETM 620 - 09U21

The normal distribution “in reverse” Example:

Given a normal distribution with μ = 40 and σ = 6, find the value of X for which 45% of the area under the normal curve is to the left of X.

1) If Φ(k) = 0.45,

k = ___________

2) Z = _______

X = _________

-5 0 5

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Normal approximation to the binomialIf n is large and p is not close to 0 or 1,

orif n is smaller but p is close to 0.5, then the binomial distribution can be approximated by the normal distribution using the transformation:

NOTE: add or subtract 0.5 from X to be sure the value of interest is included (draw a picture to know which)

Look at example 7-10, pg. 156

npq

npXZ

)5.0(

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Another exampleThe probability of a patient making a full recovery from a rare heart ailment is 0.4. If 100 people are diagnosed with this ailment, what is the probability that fewer than 30 of them recover?

p = 0.4 n = 100

μ = ____________ σ = ______________

if x = 30, then z = _____________________

and, P(X < 30) = P (Z < _________) = _________

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Your Turn Refer to the previous example,

1. What is the probability that more than 50 survive?

2. What is the probability that exactly 45 survive?

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DRAW THE PICTURE!!

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• When the random variable Y = ln(X) is normally distributed with mean μ and standard deviation σ, then X has a lognormal distribution with the density function,

Lognormal Distribution

)1(

0,2

1),;(

222

2/

])[ln(21

2

2

22

ee

e

xex

xfx

26

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ExampleThe average rate of water usage (in thousands of gallons per hour) by a certain community is know to involve the lognormal distribution with parameters μ = 5 and σ =2. What is the probability that, for any given hour, more than 50,000 gallons are used?

P(X > 50,000) = __________________________

27

Bivariate Normal distributionRandom variables [X, Y] that follow a bivariate

normal distribution have a joint density function,

and a joint probability,

This would be pretty nasty to calculate, except …X ∼ N (μX, σ2

X ) and Y ∼ N (μY , σ2Y )

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22

222

12

1exp

12

1),(

y

y

y

y

x

x

x

x

yx

yyxxyxf

1

1

2

2

),(),( 2211

b

a

b

a

dxdyyxfbYabXaP

So…It can be shown (see pp. 161-162) that,

This allows us to use the standard normal approach to solve several important problems.

ETM 620 - 09U29

xx

yy xxYE

)( 22 1( yxYVar

yy

xx yyXE

22 1( xyXVar

ExampleExample7-14, pg. 163

Given the information regarding shear strength and weld diameter in the example, what is the probability that the strength specification (1080 lbs.) can be met if the weld diameter is 0.18 in.?

ETM 620 - 09U30