Chapter 4 1 continuous random variables
This presentation is the property of its rightful owner.
Sponsored Links
1 / 99

Chapter 4-1 Continuous Random Variables PowerPoint PPT Presentation


  • 98 Views
  • Uploaded on
  • Presentation posted in: General

Chapter 4-1 Continuous Random Variables. 主講人 : 虞台文. Content. Random Variables and Distribution Functions Probability Density Functions of Continuous Random Variables The Exponential Distributions The Reliability and Failure Rate The Erlang Distributions The Gamma Distributions

Download Presentation

Chapter 4-1 Continuous Random Variables

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Chapter 4 1 continuous random variables

Chapter 4-1Continuous Random Variables

主講人:虞台文


Content

Content

  • Random Variables and Distribution Functions

  • Probability Density Functions of Continuous Random Variables

  • The Exponential Distributions

  • The Reliability and Failure Rate

  • The Erlang Distributions

  • The Gamma Distributions

  • The Gaussian or Normal Distributions

  • The Uniform Distributions


Chapter 4 1 continuous random variables1

Chapter 4-1Continuous Random Variables

Random Variables and Distribution Functions


The temperature in taipei

The Temperature in Taipei

今天中午台北市氣溫為25C之機率為何?

今天中午台北市氣溫小於或等於25C之機率為何?


Renewed definition of random variables

Renewed Definition of Random Variables

A random variable X on a probability space (, A, P) is a function

X :R

that assigns a real number X() to each sample point , such that for every real number x, the set {|X() x} is an event, i.e., a member of A.


The cumulative distribution functions

The (Cumulative) Distribution Functions

The (cumulative) distribution functionFX of a random variable X is defined to be the function

FX(x) = P(Xx), − < x < .


Example 1

Example 1


Example 11

Example 1


Example 12

R

y

Example 1


Example 13

R

y

Example 1


Example 14

Example 1


Example 15

RY

R

R/2

Example 1


Example 16

Example 1


Example 17

Example 1


Properties of distribution functions

Properties of Distribution Functions

  • 0  F(x)  1for allx;

  • Fis monotonically nondecreasing;

  • F() = 0andF() =1;

  • F(x+) = F(x)for allx.


Definition continuous random variables

Definition Continuous Random Variables

A random variable X is called a continuous random variable if


Example 2

Example 2


Chapter 4 1 continuous random variables2

Chapter 4-1Continuous Random Variables

Probability Density Functions of Continuous Random Variables


Probability density functions of continuous random variables

Probability Density Functions of Continuous Random Variables

A probability density function (pdf) fX(x) of a continuous random variable X is a nonnegative function f such that


Probability density functions of continuous random variables1

Probability Density Functions of Continuous Random Variables

A probability density function (pdf) fX(x) of a continuous random variable X is a nonnegative function f such that


Properties of pdf s

Properties of Pdf's

Remark: f(x) can be larger than 1.


Example 3

Example 3


Example 31

Example 3


Example 32

Example 3


Example 33

Example 3


Example 34

0.25926

1/3

Example 3


Chapter 4 1 continuous random variables3

Chapter 4-1Continuous Random Variables

The Exponential Distributions


The exponential distributions

The Exponential Distributions

  • The following r.v.’s are often modelled as exponential:

  • Interarrival time between two successive job arrivals.

  • Service time at a server in a queuing network.

  • Life time of a component.


The exponential distributions1

The Exponential Distributions

A r.v. X is said to possess an exponential distribution and to be exponentially distributed, denoted by

X~Exp(), if it possesses the density


The exponential distributions2

: arriving rate

: failure rate

The Exponential Distributions

pdf

cdf


The exponential distributions3

: arriving rate

: failure rate

The Exponential Distributions

pdf

cdf


Memoryless or markov property

Memoryless or Markov Property


Memoryless or markov property1

Memoryless or Markov Property


Memoryless or markov property2

Memoryless or Markov Property

Exercise:

連續型隨機變數中,唯有指數分佈具備無記憶性。


The relation between poisson and exponential distributions

: arriving rate

: failure rate

Nt

t

0

The Relation Between Poisson and Exponential Distributions

Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t].


The relation between poisson and exponential distributions1

: arriving rate

: failure rate

Nt

t

0

X

The Relation Between Poisson and Exponential Distributions

Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t].

The next arrival

Let X denote the time of the next arrival.


The relation between poisson and exponential distributions2

: arriving rate

: failure rate

Nt

X

The Relation Between Poisson and Exponential Distributions

Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t].

The next arrival

t

0

能求出P(X > t)嗎?

Let X denote the time of the next arrival.


The relation between poisson and exponential distributions3

: arriving rate

: failure rate

Nt

The Relation Between Poisson and Exponential Distributions

Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t].

The next arrival

t

0

能求出P(X > t)嗎?

X

Let X denote the time of the next arrival.


The relation between poisson and exponential distributions4

: arriving rate

: failure rate

Nt

X

The Relation Between Poisson and Exponential Distributions

Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t].

The next arrival

t

0

Let X denote the time of the next arrival.


The relation between poisson and exponential distributions5

: arriving rate

: failure rate

t1

t2

t3

t4

t5

The Relation Between Poisson and Exponential Distributions

The interarrival times of a Poisson process are exponentially distributed.


Example 5

P(“No job”) = ?

0

10 secs

Example 5

 = 0.1 job/sec


Example 51

P(“No job”) = ?

0

10 secs

Example 5

 = 0.1 job/sec

Method 1:

Let N10 represent #jobs arriving in the 10 secs.

Let X represent the time of the next arriving job.

Method 2:


Chapter 4 1 continuous random variables4

Chapter 4-1Continuous Random Variables

The Reliability

and

Failure Rate


Definition reliability

Definition  Reliability

Let r.v. X be the lifetime or time to failure of a component. The probability that the component survives until some time t is called the reliabilityR(t) of the component, i.e.,

R(t) = P(X > t) = 1  F(t)

Remarks:

  • F(t) is, hence, called unreliability.

  • R’(t)= F’(t) = f(t)is called the failure density function.


The instantaneous failure rate

The Instantaneous Failure Rate

剎那間,ㄧ切化作永恆。


The instantaneous failure rate1

t

t+t

0

t

The Instantaneous Failure Rate

生命將在時間t後瞬間結束的機率


The instantaneous failure rate2

The Instantaneous Failure Rate

生命將在時間t後瞬間結束的機率


The instantaneous failure rate3

The Instantaneous Failure Rate

瞬間暴斃率h(t)


The instantaneous failure rate4

The Instantaneous Failure Rate

瞬間暴斃率h(t)


Example 6

Example 6

Show that the failure rate of exponential distribution is characterized by a constantfailure rate.

以指數分配來model物件壽命之機率分配合理嗎?


More on failure rates

h(t)

t

More on Failure Rates

CFR


More on failure rates1

h(t)

t

More on Failure Rates

IFR

DFR

Useful Life

CFR

CFR


More on failure rates2

h(t)

t

More on Failure Rates

Exponential

Distribution

IFR

DFR

?

?

Useful Life

CFR

CFR


Relationships among f t f t r t h t

Relationships among F(t), f(t), R(t), h(t)


Relationships among f t f t r t h t1

Relationships among F(t), f(t), R(t), h(t)


Relationships among f t f t r t h t2

Relationships among F(t), f(t), R(t), h(t)


Relationships among f t f t r t h t3

Relationships among F(t), f(t), R(t), h(t)

?

?

?


C umulative h azard

Cumulative Hazard


Relationships among f t f t r t h t4

Relationships among F(t), f(t), R(t), h(t)


Example 7

Example 7


Chapter 4 1 continuous random variables5

Chapter 4-1Continuous Random Variables

The Erlang Distributions


Chapter 4 1 continuous random variables

我的老照相機與閃光燈

它只能使用四次

每使用一次後轉動九十度

使用四次後壽終正寢


The erlang distributions

time

The lifetime of my flash (X)

The Erlang Distributions

fX(t)=?

[0, )

I(X)=?


The erlang distributions1

Nt ~ P(t)

The Erlang Distributions

  • Consider a component subjected to an environment so that Nt, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t.

  • Suppose that the rth peak will cause a failure.

  • Let X denote the lifetime of the component.

  • Then,

cdf


The erlang distributions2

Nt ~ P(t)

The Erlang Distributions

Exercise of

Chapter 2

  • Consider a component subjected to an environment so that Nt, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t.

  • Suppose that the rth peak will cause a failure.

  • Let X denote the lifetime of the component.

  • Then,

pdf

cdf


The r stage erlang distributions

The r-Stage Erlang Distributions

  • Consider a component subjected to an environment so that Nt, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t.

  • Suppose that the rth peak will cause a failure.

  • Let X denote the lifetime of the component.

  • Then,

pdf

cdf


The r stage erlang distributions1

The r-Stage Erlang Distributions

pdf

cdf


The r stage erlang distributions2

The r-Stage Erlang Distributions

pdf


Example 8

Example 8

In a batch processing environment, the number of jobs arriving for service is 9 per hour. If the arrival process satisfies the requirement of a Poisson experiment. Find the probability that the elapse time between a given arrival and the fifth subsequent arrival is less than 10 minutes.

 = 9 jobs/hr.

Let X represent the time of the 5th arrival.


Chapter 4 1 continuous random variables6

Chapter 4-1Continuous Random Variables

The Gamma Distributions


Review

r為一正整數

欲將之推廣為正實數

Review

pdf


Review1

Review

pdf


The gamma distributions

The Gamma Distributions

pdf


Review2

Review


Chi square distributions

Chi-SquareDistributions


Chapter 4 1 continuous random variables7

Chapter 4-1Continuous Random Variables

The Gaussian or Normal Distributions


The gaussian or normal distributions

The Gaussian or Normal Distributions

德國的10馬克紙幣, 以高斯(Gauss, 1777-1855)為人像, 人像左側有一常態分佈之p.d.f.及其圖形。


The gaussian or normal distributions1

The Gaussian or Normal Distributions

pdf


The gaussian or normal distributions2

 : mean

 : standard deviation

2: variance

The Gaussian or Normal Distributions

Inflection

point

Inflection

point


The gaussian or normal distributions3

 : mean

 : standard deviation

2: variance

The Gaussian or Normal Distributions

varying

varying


The gaussian or normal distributions4

 : mean

 : standard deviation

2: variance

The Gaussian or Normal Distributions

Facts:


The gaussian or normal distributions5

 : mean

 : standard deviation

2: variance

The Gaussian or Normal Distributions


Standard normal distribution

Standard Normal Distribution


Table of n 0 1

Table of N(0, 1)

z


Table of n 0 11

Table of N(0, 1)

z

Fact:


Probability evaluation for n 2

x

Probability Evaluation for N(, 2)


Probability evaluation for n 21

x

Probability Evaluation for N(, 2)


Probability evaluation for n 22

x

Fact:

Probability Evaluation for N(, 2)

Z-Score:表距離中心若干個標準差


Example 9

Example 9

X ~ N(12.00, 0.202)


Example 91

X ~ N(12.00, 0.202)

Example 9


Example 92

X ~ N(12.00, 0.202)

Example 9


Example 93

X ~ N(12.00, 0.202)

Example 9


Example 10

|X | < 

|X | < 2

|X | < 3

Example 10


Example 101

Example 10


Example 102

Example 10


Example 103

Example 10


Chapter 4 1 continuous random variables8

Chapter 4-1Continuous Random Variables

The Uniform Distributions


The uniform distributions

f(x)

x

a

b

F(x)

1

x

a

b

The Uniform Distributions

pdf

cdf


Summary

Summary

  • The Exponential Distributions

  • The Erlang Distributions

  • The Gamma Distributions

  • The Gaussian or Normal Distributions

  • The Uniform Distributions


  • Login