[PDF] - Outline Continuous Probability Distributions The Uniform PDF Document

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12/14/2006 1

219323 Probability y and Statistics for Software and Knowledge Engineers

Lecture 5: Continuous Probability Distributions

Monchai Sopitkamon, Ph.D.

Outline

Continuous Probability Distributions

– The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5)

Normal Distribution Normal Distribution

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Continuous Probability Distributions: The Uniform Distribution I (4.1)

A RV X w/ a flat pdf between two

A RV X w/ a flat pdf between two points, a and b, so that

for a ≤ x ≤ b and f(x) = 0 elsewhere

X b itt X U( b) a b x f − = 1 ) (

X can be written as X ∼ U(a, b) The cdf is for a ≤ x ≤ b

a b a x x F − − = ) (

2 ) ( b a X E + =

12 ) ( ) (

2

a b X Var − =

Continuous Probability Distributions: The Uniform Distribution II (4.1)

Figure 4 .1 Probability density function of a

U( a, b) distribution

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Continuous Probability Distributions: The Uniform Distribution III (4.1)

Ex.30 pg.186

SD (σ) = √8 33 = 2 89 mm (cont )

33 . 8 12 ) 10 ( ) (

2

= − = X Var

SD (σ) = √8.33 = 2.89 mm … (cont.)

Outline

Continuous Probability Distributions

– The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5)

Normal Distribution Normal Distribution

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Continuous Probability Distributions: The Exponential Distribution I (4.2)

Used to model failure or waiting times

Used to model failure or waiting times and interarrival times

The pdf is for x ≥ 0, λ > 0

and f(x) = 0 for x < 0

The cdf is for x ≥ 0

x

e x f

λ

−

= ) (

x

e x F

λ −

− =1 ) ( and λ 1 ) ( = X E

2

1 ) ( λ = X Var

Continuous Probability Distributions: The Exponential Distribution II (4.2)

Figure 4 .3 pdf of an exponential Figure 4 .4 pdf of an exponential p p distribution w ith λ = 1 p p distribution w ith λ = 1 / 2

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12/14/2006 5 Continuous Probability Distributions: The Memoryless Property of the Exponential Distribution I (4.2.2)

Given that P(X ≥ x) = 1 – F(x) = e-λx

Then if the RV Y represents the Then, if the RV Y represents the additional time beyond x0 that elapses before the event occurs, P(Y ≥ y) = P(X ≥ x0 + y | X ≥ x0)

y y x

e e y x X P

λ λ − + −

= = + ≥ =

) (

) ( so that Y also has an exponential distribution with parameter λ

y x

e e x X P

λ −

= = ≥ =

0)

(

Continuous Probability Distributions: The Memoryless Property of the Exponential Distribution II (4.2.2)

Figure 4 .5 I llustration of the m em oryless property of the exponential property of the exponential

distribution. The part of the

probability density function beyond x 0 is a scaled version of the w hole probability density function

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12/14/2006 6 Continuous Probability Distributions: The Memoryless Property Implications (4.2.2)

Exponential distribution is the only

continuous distribution with the memoryless t property

Thus exponential distribution is suitable for

modeling failure or waiting time where time elapses have no effect on the prob, e.g., to model the failure time of an electronic part that fails due to voltage fluctuation.

However, if the failure is due to wear out,

then the memoryless property is not then the memoryless property is not plausible and exponential distribution should not be used to model the failure times. Continuous Probability Distributions: The Poisson Process I (4.2.3)

A Poisson process w/ parameter λ is

a process where the distributions of p the time intervals between the

ccurrences of adjacent events are

independent RVs having exponential distributions w/ parameters λ

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12/14/2006 7 Continuous Probability Distributions: The Poisson Process II (4.2.3)

Can be used to model

th i l f ll t it hb d – the arrival of calls at a switchboard – the addition of new elements to a queue – the arrival of network packets to a switch

The expected waiting time between The expected waiting time between

two events in a Poisson process = the expected value of an exponential distribution w/ parameter λ = 1/ λ

Continuous Probability Distributions: The Poisson Process III (4.2.3)

If RV X counts the number of events

ccurring within a fixed time interval
f length t, then X ∼ P(λt.)

! ) ( ) ( x t e x X P

x t λ λ −

= =

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Continuous Probability Distributions: Examples of the Exponential Distribution (4.2.4)

Ex.31, pg.192

Figure 4 .8 Probability density function for shipw reck hunt

(see Excel Spreadsheet)

Outline

Continuous Probability Distributions

– The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5)

Normal Distribution Normal Distribution

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Continuous Probability Distributions: The Gamma Distribution I (4.3)

Used in the area of reliability theory and

analysis of Poisson process

The Gamma Function The Gamma Function

for k > 0 Note that Γ(1) = 1 and Γ(1/2) = √π, and i l

∫

∞ − −

= Γ

1

) ( dx e x k

x k

( ) ( ) in general, Γ(k) = (k – 1) Γ(k – 1) for k > 1 If n is a positive integer, then Γ(n) = (n – 1)!

Continuous Probability Distributions: The Gamma Distribution II (4.3)

The Gamma Distribution w/

The Gamma Distribution w/ parameter k > 0 and λ > 0 has a pdf for x ≥ 0 and f(x) = 0 for x < 0

) ( ) (

1

k e x x f

x k k

Γ =

− − λ

λ

k: shape parameter λ: scale parameter

Expectation Variance

λ k X E = ) (

2

) ( λ k X Var =

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Continuous Probability Distributions: The Gamma Distribution III (4.3)

Figure 4 .1 3 Probability density functions of gam m a distributions

Continuous Probability Distributions: The Gamma Distribution IV (4.3)

If X1, …, Xk are independent RVs

each having an exponential dist w/ each having an exponential dist w/ parameter λ, then the RV X = X1+ …+ Xk has a gamma dist w/ parameters k and λ Thi i li th t f P i

This implies that for a Poisson

process w/ parameter λ, the time taken for k events to occur has a gamma dist w/ parameters k and λ

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Continuous Probability Distributions: The Gamma Distribution V (4.3)

Ex.32 pg.200:

m k X E 16 . 1 3 4 5 ) ( = = = λ 3 . 4 λ

Figure 4 .1 5 Distance to fifth fracture has a

gam m a distribution w ith param eters k = 5 and λ = 4 .3 (see Excel spreadsheet)

Outline

Continuous Probability Distributions

– The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5)

Normal Distribution Normal Distribution

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Continuous Probability Distributions: The Weibull Distribution I (4.4)

Used to model failure and waiting

times

shape parameter scale parameter

The pdf is

with parameters a > 0 and λ > 0, for x ≥ 0 and f(x) = 0 for x < 0

The cdf is

f

a

x a a

e x a x f

) ( 1

) (

λ

− −

=

a

x

e x F

) (

1 ) (

λ −

− = for x ≥ 0

Expectation Variance

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Γ = a X E 1 1 1 ) ( λ ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Γ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Γ =

2 2

1 1 2 1 1 ) ( a a X Var λ

Continuous Probability Distributions: The Weibull Distribution II (4.4)

Figure 4.17 Probability density functions of the Weibull distribution

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Continuous Probability Distributions: The Weibull Distribution III (4.4)

Ex.33 pg.204:

Figure 4 1 9 Figure 4 .1 9

Distribution of bacteria survival tim es

(see Excel spreadsheet)

Outline

Continuous Probability Distributions

– The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5)

Normal Distribution Normal Distribution

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Continuous Probability Distributions: The Beta Distribution I (4.5)

Used to model proportions The pdf

when a > 0 and b > 0 are parameters for 0 ≤ x ≤ 1, and f(x) = 0 elsewhere E i

1 1

) 1 ( ) ( ) ( ) ( ) (

− −

− Γ Γ + Γ =

b a

x x b a b a x f a

Expectation Variance

b a a X E + = ) (

) 1 ( ) ( ) (

2

+ + + = b a b a ab X Var

Continuous Probability Distributions: The Beta Distribution II (4.5)

Figure 4 .2 2 Probability density functions of the

beta distribution

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Continuous Probability Distributions: The Beta Distribution II (4.5)

Ex. 35 pg.207: