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Chapter 4. Continuous Probability Distributions

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## Chapter 4. Continuous Probability Distributions

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**Chapter 4. Continuous Probability Distributions**4.1 The Uniform Distribution 4.2 The Exponential Distribution 4.3 The Gamma Distribution 4.4 The Weibull Distribution 4.5 The Beta Distribution**4.1 The Uniform Distribution4.1.1 Definition of the Uniform**Distribution(1/2) • It has a flat pdf over a region. • if X takes on values between a and b, and • Mean and Variance**Figure 4.1 Probability density function of aU(a, b)**distribution**4.2 The Exponential Distribution4.2.1 Definition of the**Exponential Distribution • Pdf: • Cdf: • Mean and variance:**Figure 4.3Probability density function of an exponential**distribution with parameter l= 1**The exponential distribution often arises, in practice, as**being of the amount of time until some specific event occurs. For example, the amount of time until an earthquake occurs, the amount of time until a new war breaks out, or the amount of time until a telephone call you receive turns out to be a wrong number, etc.**4.2.2 The memoryless property of the Exponential**Distribution • For any non-negative x and y • The exponential distribution is the only continuous distribution that has the memoryless property**Memoryless property of exponential random variable:**This is equivalent to When X is an exponential random variable, The memoryless condition is satisfied since**Example: Suppose that a number of miles that a car run**before its battery wears out is exponentially distributed with an average value of 10,000 miles. If a person desires to take a 5,000 mile trip, what is the probability that he will be able to complete his trip without to replace the battery? (Sol) Let X be a random variable including the remaining lifetime (in thousand miles) of the battery. Then, What if X is not exponential random variable? That is, an additional information of t should be known.**Proposition: If are independent**exponential random variables having respective parameters , then is the exponential random variable with parameter (proof)**Example: A series system is one that all of its components**to function in order for the system itself to be functional. For an n component series system in which the component lifetimes are independent exponential random variables with respective parameters . What is the probability the system serves for a time t? (Sol) Let X be a random variable indicating the system lifetime. Then, X is an exponential random variable with parameter Hence,**4.2.3 The Poisson process**• A stochastic process is a sequence of random events • A Poisson process with parameter is a stochastic process where the time (or space) intervals between event-occurrences follow the Exponential distribution with parameter . • If X is the number of events occurring within a fixed time (or space) interval of length t, then**Figure 4.7 A Poisson process. The number of events**occurring in a time interval of length t has a Poisson distribution with mean lt**The Poisson Process**Suppose that events are occurring at random time points and let N(t) denote the number of points that occur in time interval [0,t]. Then, A Poisson process having rate is defined if (a) N(0)=0, (b) the number of events that occur in disjoint time intervals are independent, (c) the distribution of N(t) depends only on the length of interval, (d) and (e)**Let us break the interval [0,t] into n non-overlapping**subintervals each of length t/n. Now, there will be k events in [0,t] if either (1) N(t) equals to k and there is at most one event in each subinterval, or (2) N(t) equals to k and at least one of subintervals contain 2 or more events. Then, Since the number of events in the different subintervals are independent, it follows that**Hence, as n approach infinity,**That is, the number of events in any interval of length t has a Poisson distribution with mean For a Poisson process, let denote the time of the first event and for n>1, denote the elapsed time between the (n-1)st and nth event. Then, the sequence is called the sequence of inter-arrival times.**The distribution of**The event takes place if and only if no events of the Poisson process occur in [0,t] and thus, Likewise, Repeating the same argument yields are independent exponential random variables with mean**Example 32 (Steel Girder Fractures, p.209)**• 42 fractures on average on a 10m long girder between-fracture length: 10/43=0.23m on average • If the between-fracture length (X) follows an exponential distribution, how would you define the gap? • How would you define the number of fractures (Y) per 1m steel girder? • How are the Exponential distribution and the Poisson distribution related?**Figure 4.9 Poisson process modeling fracturelocations on a**steel girder**P(the length of a gap is less than 10cm)=?**• P(a 25-cm segment of a girder contains at least two fractures)=?**Figure 4.10 The number of fractures in a 25-cm segment of**the steel girder has a Poisson distribution with mean1.075**4.3 The Gamma Distribution4.3.1 Definition of the Gamma**distribution • Useful for reliability theory and life-testing and has several important sister distributions • The Gamma function: • The Gamma pdf with parameters k>0 and >0: • Mean and variance:**Calculation of Gamma function**When k is an integer, When k=1, the gamma distribution is reduced to the exponential with mean**Properties of Gamma random variables**If are independent gamma random variables with respective parameters then is a gamma random variable with parameters • The gamma random variable with parameters is equivalent to the exponential random variable with parameter • If are independent exponential random variables, each having rate , then is a gamma random variable with parameters**4.3.2 Examples of the Gamma distribution**• Example 32 (Steel Girder Fractures) : • Y= the number of fractures within 1m of the girder**Figure 4.15 Distance to fifth fracture has agamma**distribution with parameters k = 5 and l= 4.3**4.4 Weibull distribution4.4.1 Definition of the Weibull**distribution • Useful for modeling failure and waiting times (see Examples 33 & 34) • The pdf: • Mean and variance:**The c.d.f. of Weibull distribution:**The pth quantile of Weibull distribution: Find x such that**4.5 The Beta distribution**• Useful for modeling proportions and personal probability (See Examples 35 & 36.) • Pdf: • Mean and variance:**Figure 4.21 Probability density functions of the beta**distribution**Figure 4.22 Probability density functions of the beta**distribution**Exponential and Weibull random variables have as**their set of possible values. • In engineering applications of probability theory, it is occasionally helpful to have available family of distributions whose set of possible values is finite interval. • One of such family is the beta family of distributions.**Example: the beta distribution and rainstorms.**Data gathered by the U.S. Weather Service in Alberquerque, concern the fraction of the total rainfall falling during the first 5 minutes of storms occurring during both summer and nonsummer seasons. The data for 14 nonsummer storms can be described reasonably well by a standard beta distribution with a=2.0 and b=8.8. • Let X be the fraction of the storm’s rainfall falling during the first 5 minutes. Then, the probability that more than 20% of the storm’s rainfall during the first 5 minutes is determined by