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# Chapter 4-1 Continuous Random Variables - PowerPoint PPT Presentation

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

### 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

### 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 < .

R

y

R

y

RY

R

R/2

### 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

## 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.

0.25926

1/3

## 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

pdf

cdf

: arriving rate

: failure rate

pdf

cdf

Exercise:

: 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

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

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

 = 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.

t

t+t

0

t

### Example 6

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

h(t)

t

CFR

h(t)

t

IFR

DFR

Useful Life

CFR

CFR

h(t)

t

Exponential

Distribution

IFR

DFR

?

?

Useful Life

CFR

CFR

?

?

?

## Chapter 4-1Continuous Random Variables

The Erlang Distributions

### 我的老照相機與閃光燈

time

The lifetime of my flash (X)

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

pdf

cdf

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

pdf

pdf

pdf

## Chapter 4-1Continuous Random Variables

The Gaussian or Normal Distributions

### 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

z

z

Fact:

x

x

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)

|X | < 

|X | < 2

|X | < 3

## Chapter 4-1Continuous Random Variables

The Uniform Distributions

f(x)

x

a

b

F(x)

1

x

a

b

pdf

cdf

### Summary

• The Exponential Distributions

• The Erlang Distributions

• The Gamma Distributions

• The Gaussian or Normal Distributions

• The Uniform Distributions