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

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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

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Figure 4.1 Probability density function of
aU(a, b) distribution
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4.2 The Exponential Distribution4.2.1 Definition
of the Exponential Distribution

• Pdf
• Cdf
• Mean and variance

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Figure 4.3Probability density function of an
exponential distribution with parameter l 1
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• 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.

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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

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• Memoryless property of exponential random
  variable
• This is equivalent to
• When X is an exponential random variable,
• The memoryless condition is satisfied since

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• 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.

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• Proposition If are
  independent exponential random variables having
  respective parameters , then
  is the exponential random
  variable with
• parameter
• (proof)

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• 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,

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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

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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
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• 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)

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• 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

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• 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 ngt1, denote
  the elapsed time between the (n-1)st and nth
  event. Then, the sequence
• is called the sequence of inter-arrival
  times.

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• 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

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• Example 32 (Steel Girder Fractures, p.209)
• 42 fractures on average on a 10m long girder
• between-fracture length
• 10/430.23m 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?

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Figure 4.9 Poisson process modeling
fracturelocations on a steel girder
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• P(the length of a gap is less than 10cm)?
• P(a 25-cm segment of a girder contains at least
  two fractures)?

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Figure 4.10 The number of fractures in a 25-cm
segment of the steel girder has a Poisson
distribution with mean 1.075
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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 kgt0 and gt0
• Mean and variance

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Curves of Gamma pdf
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• Calculation of Gamma function
• When k is an integer,
• When k1, the gamma distribution is reduced to
  the exponential
• with mean

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• 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

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4.3.2 Examples of the Gamma distribution

• Example 32 (Steel Girder Fractures)
• Y the number of fractures within 1m of the
  girder

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Figure 4.15 Distance to fifth fracture has
agamma distribution with parameters k 5 and l
4.3
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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

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Curves of the Weibull distribution
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• The c.d.f. of Weibull distribution
• The pth quantile of Weibull distribution
• Find x such that

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4.5 The Beta distribution

• Useful for modeling proportions and personal
  probability
• (See Examples 35 36.)
• Pdf
• Mean and variance

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Figure 4.21 Probability density functions of the
beta distribution
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Figure 4.22 Probability density functions of the
beta distribution
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• 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.

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• 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 a2.0 and b8.8.
• Let X be the fraction of the storms rainfall
  falling during the first 5 minutes. Then, the
  probability that more than 20 of the storms
  rainfall during the first 5 minutes is determined
  by