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
Chapter 4-1 Continuous Random Variables Random Variables and Distribution Functions
The Temperature in Taipei 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?
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 FX of a random variable X is defined to be the function FX(x) = P(X x), − < x < .
Example 1
Example 1
Example 1 R y
Example 1 R y
Example 1
Example 1 RY R R/2
Example 1
Example 1
Properties of Distribution Functions 0 F(x) 1 for all x; F is monotonically nondecreasing; F() = 0 and F() =1; F(x+) = F(x) for all x.
Definition Continuous Random Variables A random variable X is called a continuous random variable if
Example 2
Chapter 4-1 Continuous 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
Example 3 0.25926 1/3
Chapter 4-1 Continuous 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
The Exponential Distributions : arriving rate : failure rate The Exponential Distributions pdf cdf
The Exponential Distributions : 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。
The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt t
The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt The next arrival t X Let X denote the time of the next arrival.
The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt The next arrival t 能求出P(X > t)嗎? X Let X denote the time of the next arrival.
The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt The next arrival t 能求出P(X > t)嗎? X Let X denote the time of the next arrival.
The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt The next arrival t X Let X denote the time of the next arrival.
The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate t1 t2 t3 t4 t5 The interarrival times of a Poisson process are exponentially distributed.
P(“No job”) = ? 10 secs Example 5 = 0.1 job/sec
Example 5 = 0.1 job/sec Method 1: P(“No job”) = ? 10 secs Example 5 = 0.1 job/sec Method 1: Let N10 represent #jobs arriving in the 10 secs. Method 2: Let X represent the time of the next arriving job.
Chapter 4-1 Continuous 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 reliability R(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 t+t t
The Instantaneous Failure Rate
The Instantaneous Failure Rate 瞬間暴斃率h(t)
The Instantaneous Failure Rate 瞬間暴斃率h(t)
Example 6 以指數分配來model物件壽命之機率分配合理嗎? 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) CFR
More on Failure Rates t h(t) DFR IFR Useful Life CFR CFR
? ? More on Failure Rates Exponential Distribution h(t) t DFR IFR ? ? 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 4-1 Continuous 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
The Erlang Distributions time The lifetime of my flash (X) [0, ) fX(t)=? I(X)=?
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
The Erlang Distributions 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 = 9 jobs/hr. Let X represent the time of the 5th arrival. 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 4-1 Continuous 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-Square Distributions
Chapter 4-1 Continuous 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
The Gaussian or Normal Distributions : mean : standard deviation 2: variance The Gaussian or Normal Distributions Inflection point Inflection point
The Gaussian or Normal Distributions : mean : standard deviation 2: variance The Gaussian or Normal Distributions varying varying
The Gaussian or Normal Distributions : mean : standard deviation 2: variance The Gaussian or Normal Distributions Facts:
The Gaussian or Normal Distributions : 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:
Probability Evaluation for N(, 2) x
Probability Evaluation for N(, 2) x
Probability Evaluation for N(, 2) Fact: Probability Evaluation for N(, 2) x 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
Example 10 |X | < |X | < 2 |X | < 3
Example 10
Example 10
Example 10
Chapter 4-1 Continuous Random Variables The Uniform Distributions
The Uniform Distributions a b x f(x) pdf a b x F(x) 1 cdf
Summary The Exponential Distributions The Erlang Distributions The Gamma Distributions The Gaussian or Normal Distributions The Uniform Distributions