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Chapter 4-1 Continuous Random Variables

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## Chapter 4-1 Continuous Random Variables

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**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**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 functionFX of a random variable X is defined to be the function FX(x) = P(Xx), − < x < .**R**y Example 1**R**y Example 1**RY**R R/2 Example 1**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 Example 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 The Exponential Distributions pdf cdf**: arriving rate**: failure rate The Exponential Distributions 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。**: 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 能求出P(X > t)嗎? 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 能求出P(X > t)嗎? 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 Example 5 = 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.**The Instantaneous Failure Rate**剎那間，ㄧ切化作永恆。**t**t+t 0 t The Instantaneous Failure Rate 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 constantfailure rate. 以指數分配來model物件壽命之機率分配合理嗎?