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Chapter 4-1 Continuous Random VariablesPowerPoint Presentation

Chapter 4-1 Continuous Random Variables

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### Chapter 4-1Continuous Random Variables

### Chapter 4-1Continuous Random Variables

### Chapter 4-1Continuous Random Variables

### Chapter 4-1Continuous Random Variables

### Chapter 4-1Continuous Random Variables

pdf

### Chapter 4-1Continuous Random Variables

### Chapter 4-1Continuous Random Variables

### 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

Random Variables and Distribution Functions

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(Xx), − < x < .

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

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.

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

: failure rate

Nt

t

0

The Relation Between Poisson and Exponential DistributionsLet r.v. Nt denote #jobs arriving to a computer system in the interval (0, t].

: failure rate

Nt

t

0

X

The Relation Between Poisson and Exponential DistributionsLet 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.

: failure rate

Nt

X

The Relation Between Poisson and Exponential DistributionsLet 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.

: failure rate

Nt

The Relation Between Poisson and Exponential DistributionsLet 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.

: failure rate

Nt

X

The Relation Between Poisson and Exponential DistributionsLet 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.

: failure rate

t1

t2

t3

t4

t5

The Relation Between Poisson and Exponential DistributionsThe interarrival times of a Poisson process are exponentially distributed.

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:

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。

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?

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)

The Erlang Distributions

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 DistributionsExercise 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,

cdf

The r-Stage Erlang Distributions pdf

- 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

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.

The Gamma Distributions

Chi-SquareDistributions

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。

: mean

: standard deviation

2: variance

The Gaussian or Normal DistributionsInflection

point

Inflection

point

Example 9

X ~ N(12.00, 0.202)

X ~ N(12.00, 0.202)

Example 9X ~ N(12.00, 0.202)

Example 9X ~ N(12.00, 0.202)

Example 9The Uniform Distributions

Summary

- The Exponential Distributions
- The Erlang Distributions
- The Gamma Distributions
- The Gaussian or Normal Distributions
- The Uniform Distributions

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