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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

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Chapter 4-1 Continuous Random Variables

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Chapter 4-1Continuous 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

  • The Gaussian or Normal Distributions

  • The Uniform Distributions


Chapter 4-1Continuous Random Variables

Random Variables and Distribution Functions


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

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 functionFX of a random variable X is defined to be the function

FX(x) = P(Xx), − < x < .


Example 1


Example 1


R

y

Example 1


R

y

Example 1


Example 1


RY

R

R/2

Example 1


Example 1


Example 1


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

A random variable X is called a continuous random variable if


Example 2


Chapter 4-1Continuous 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 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

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


Example 3


Example 3


Example 3


Example 3


0.25926

1/3

Example 3


Chapter 4-1Continuous Random Variables

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 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


: arriving rate

: failure rate

The Exponential Distributions

pdf

cdf


: arriving rate

: failure rate

The Exponential Distributions

pdf

cdf


Memoryless or Markov Property


Memoryless or Markov Property


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。


: 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].


: 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.


: 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.


: 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.


: 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.


: 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.


P(“No job”) = ?

0

10 secs

Example 5

 = 0.1 job/sec


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 4-1Continuous Random Variables

The Reliability

and

Failure Rate


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

剎那間,ㄧ切化作永恆。


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 Rate

生命將在時間t後瞬間結束的機率


The Instantaneous Failure Rate

瞬間暴斃率h(t)


The Instantaneous Failure Rate

瞬間暴斃率h(t)


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?


h(t)

t

More on Failure Rates

CFR


h(t)

t

More on Failure Rates

IFR

DFR

Useful Life

CFR

CFR


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(t)


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

?

?

?


Cumulative Hazard


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


Example 7


Chapter 4-1Continuous Random Variables

The Erlang Distributions


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


time

The lifetime of my flash (X)

The Erlang Distributions

fX(t)=?

[0, )

I(X)=?


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


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

  • 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

pdf

cdf


The r-Stage Erlang Distributions

pdf


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 4-1Continuous Random Variables

The Gamma Distributions


r為一正整數

欲將之推廣為正實數

Review

pdf


Review

pdf


The Gamma Distributions

pdf


Review


Chi-SquareDistributions


Chapter 4-1Continuous Random Variables

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 Distributions

pdf


 : mean

 : standard deviation

2: variance

The Gaussian or Normal Distributions

Inflection

point

Inflection

point


 : mean

 : standard deviation

2: variance

The Gaussian or Normal Distributions

varying

varying


 : mean

 : standard deviation

2: variance

The Gaussian or Normal Distributions

Facts:


 : mean

 : standard deviation

2: variance

The Gaussian or Normal Distributions


Standard Normal Distribution


Table of N(0, 1)

z


Table of N(0, 1)

z

Fact:


x

Probability Evaluation for N(, 2)


x

Probability Evaluation for N(, 2)


x

Fact:

Probability Evaluation for N(, 2)

Z-Score:表距離中心若干個標準差


Example 9

X ~ N(12.00, 0.202)


X ~ N(12.00, 0.202)

Example 9


X ~ N(12.00, 0.202)

Example 9


X ~ N(12.00, 0.202)

Example 9


|X | < 

|X | < 2

|X | < 3

Example 10


Example 10


Example 10


Example 10


Chapter 4-1Continuous Random Variables

The Uniform Distributions


f(x)

x

a

b

F(x)

1

x

a

b

The Uniform Distributions

pdf

cdf


Summary

  • The Exponential Distributions

  • The Erlang Distributions

  • The Gamma Distributions

  • The Gaussian or Normal Distributions

  • The Uniform Distributions


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