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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

- 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

今天中午台北市氣溫為25C之機率為何?

今天中午台北市氣溫小於或等於25C之機率為何?

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

FX(x) = P(Xx), − < x < .

R

y

R

y

RY

R

R/2

- 0 F(x) 1for allx;
- Fis monotonically nondecreasing;
- F() = 0andF() =1;
- F(x+) = F(x)for allx.

A random variable X is called a continuous random variable if

Chapter 4-1Continuous 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

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

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

0.25926

1/3

Chapter 4-1Continuous Random Variables

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.

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

cdf

: arriving rate

: failure rate

cdf

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

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

: arriving rate

: failure rate

Nt

t

0

X

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

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

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

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 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

= 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

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.

剎那間，ㄧ切化作永恆。

t

t+t

0

t

生命將在時間t後瞬間結束的機率

生命將在時間t後瞬間結束的機率

瞬間暴斃率h(t)

瞬間暴斃率h(t)

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

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

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)

fX(t)=?

[0, )

I(X)=?

Nt ~ P(t)

- 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)

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,

cdf

- 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

cdf

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

Chapter 4-1Continuous Random Variables

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

Inflection

point

Inflection

point

: mean

: standard deviation

2: variance

varying

varying

: mean

: standard deviation

2: variance

Facts:

: mean

: standard deviation

2: variance

z

z

Fact:

x

x

x

Fact:

Z-Score:表距離中心若干個標準差

X ~ N(12.00, 0.202)

X ~ N(12.00, 0.202)

X ~ N(12.00, 0.202)

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

cdf

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