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 The Gaussian or Normal Distributions The Uniform Distributions

Random Variables and Distribution Functions Chapter 4-1 Continuous Random Variables

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 function F X of a random variable X is defined to be the function F X (x) = P(X  x), −  < x < .

Example 1

R y

R y

R RYRY R/2

Example 1

Properties of Distribution Functions 1. 0  F(x)  1 for all x ; 2. F is monotonically nondecreasing; 3. F(  ) = 0 and F(  ) =1 ; 4. F(x+) = F(x) for all x.

Definition  Continuous Random Variables A random variable X is called a continuous random variable if

Example 2

Probability Density Functions of Continuous Random Variables Chapter 4-1 Continuous Random Variables

Probability Density Functions of Continuous Random Variables A probability density function (pdf) f X (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) f X (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

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The Exponential Distributions Chapter 4-1 Continuous Random Variables

The Exponential Distributions The following r.v.’s are often modelled as exponential: 1. Interarrival time between two successive job arrivals. 2. Service time at a server in a queuing network. 3. 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

The Exponential Distributions : arriving rate : failure rate pdf cdf

The Exponential Distributions : arriving rate : failure rate pdf cdf

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

The Relation Between Poisson and Exponential Distributions Let X denote the time of the next arrival. : arriving rate : failure rate The next arrival NtNt Let r.v. N t denote #jobs arriving to a computer system in the interval (0, t]. t 0 X

The Relation Between Poisson and Exponential Distributions Let X denote the time of the next arrival. : arriving rate : failure rate The next arrival NtNt Let r.v. N t denote #jobs arriving to a computer system in the interval (0, t]. t 0 X 能求出 P(X > t) 嗎 ?

The Relation Between Poisson and Exponential Distributions Let X denote the time of the next arrival. : arriving rate : failure rate The next arrival NtNt Let r.v. N t denote #jobs arriving to a computer system in the interval (0, t]. t 0 X 能求出 P(X > t) 嗎 ?

The Relation Between Poisson and Exponential Distributions Let X denote the time of the next arrival. : arriving rate : failure rate NtNt Let r.v. N t denote #jobs arriving to a computer system in the interval (0, t]. t 0 X The next arrival

The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate t1t1 t2t2 t3t3 t4t4 t5t5 The interarrival times of a Poisson process are exponentially distributed.

Example secs = 0.1 job/sec P (“ No job ” ) = ?

Example secs = 0.1 job/sec P (“ No job ” ) = ? Method 1: Method 2: Let N 10 represent #jobs arriving in the 10 secs. Let X represent the time of the next arriving job.

The Reliability and Failure Rate Chapter 4-1 Continuous Random Variables

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 reliability R(t) of the component, i.e., R(t) = P(X > t) = 1  F(t) Remarks: 1. F(t) is, hence, called unreliability. 2. 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 0t tt t+  t 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 constant failure rate. 以指數分配來 model 物件壽命之機率分配合理嗎 ?

More on Failure Rates t h(t)h(t) CFR

More on Failure Rates t h(t)h(t) CFR Useful Life CFR DFR IFR

More on Failure Rates t h(t)h(t) CFR Useful Life CFR DFR IFR Exponential Distribution Exponential Distribution ??

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

? ? ?

Cumulative Hazard

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

Example 7

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 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 Erlang Distributions The lifetime of my flash ( X ) I(X)=? f X (t)=? [0,  )

The Erlang Distributions Consider a component subjected to an environment so that N t, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t. Suppose that the r th peak will cause a failure. Let X denote the lifetime of the component. Then, N t ~ P( t) cdf

The Erlang Distributions Consider a component subjected to an environment so that N t, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t. Suppose that the r th peak will cause a failure. Let X denote the lifetime of the component. Then, N t ~ P( t) Exercise of Chapter 2 cdf pdf

The r-Stage Erlang Distributions Consider a component subjected to an environment so that N t, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t. Suppose that the r th peak will cause a failure. Let X denote the lifetime of the component. Then, cdf pdf

The r-Stage Erlang Distributions cdf pdf

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. Let X represent the time of the 5 th arrival. = 9 jobs/hr.

The Gamma Distributions Chapter 4-1 Continuous Random Variables

Review pdf r 為一正整數 欲將之推廣為正實數 r 為一正整數 欲將之推廣為正實數

Review pdf   

The Gamma Distributions pdf

Review

Chi-Square Distributions

The Gaussian or Normal Distributions Chapter 4-1 Continuous Random Variables

The Gaussian or Normal Distributions 德國的 10 馬克紙幣, 以高斯 (Gauss, ) 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

The Gaussian or Normal Distributions  : mean  : standard deviation  2 : variance Inflection point Inflection point

The Gaussian or Normal Distributions  : mean  : standard deviation  2 : variance varying  varying 

The Gaussian or Normal Distributions  : mean  : standard deviation  2 : variance Facts:

The Gaussian or Normal Distributions  : mean  : standard deviation  2 : variance

Standard Normal Distribution

Table of N(0, 1) z

z Fact:

Probability Evaluation for N( ,  2 ) x 

x 

x  Fact:

Example 9 X ~ N(12.00, )

Example 9 X ~ N(12.00, )

Example 9 X ~ N(12.00, )

Example 9 X ~ N(12.00, )

Example 10  |X   | <  |X   | < 2  |X   | < 3 

Example 10

The Uniform Distributions Chapter 4-1 Continuous Random Variables

The Uniform Distributions pdf cdf ab x f(x)f(x) ab x F(x)F(x) 1

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