010'141 Engineering Mathematics II Lecture 4 Continuous Distributions

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010.141 Engineering Mathematics IILecture
4Continuous Distributions

• Bob McKay
• School of Computer Science and Engineering
• College of Engineering
• Seoul National University
• Partly based on
• Sheldon Ross A First Course in Probability

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Outline

• Continuous Random Variables
• Continuous Distributions
• Uniform
• Normal
• Exponential
• Other Discrete Distributions
• Gamma
• Weibull
• Cauchy
• Beta
• Functions of a Random Variable

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Continuous Random Variables

• Discrete random variables cover many important
  problems, but there are also many problems which
  cannot be conveniently represented in this way
• Recall that a random variable over a probability
  space (?, ?,P) is a measurable function X ? ? S
  for some set S
• If ? is a continuous space - for example R, the
  real numbers, we call X a continuous random
  variable
• In the case of R, this is equivalent to an
  alternative definition, that there exists a
  non-negative function f defined over R, such that
  for any measurable B ? R,
• PX ?B ?B f(x) dx
• f is known as the probability density function

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Properties ofContinuous Random Variables

• Of course, for the case B R
• ?R f(x) dx PX ? R 1
• For intervals of R, we have the equivalent
• PX ? a,b ?ab f(x) dx
• So PX ? a,a ?aa f(x) dx 0
• That is, the probability of any specific value is
  zero
• PX lt a ?-?a f(x) dx

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Continuous Example (Ross)

• The amount of time, in hours, that a computer
  functions before breaking down is a continuous
  random variable with probability density function
  given by
• f(x) ?e-x/100 x ? 0
• 0 x lt 0
• What is the probability that a computer will
  function between 50 and 150 hours before breaking
  down?

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Continuous Example

• We have
• ?-?? f(x) dx ? ?0? e-x/100 dx 1
• From which
• ? 1/100
• So
• PX ? 50,150 ?50150 1/100 e-x/100 dx
  e-1/2 - e-3/2 0.633

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Uniform Distribution

• The probability density function
• f(x) 1 0 lt x lt 1 0 otherwise
• generates a uniform distribution over the
  interval (0,1)
• That is, any value between 0 and 1 is equally
  likely
• This may be generalised to an arbitrary interval
  (a,b)
• f(x) 1/b-a a lt x lt b 0 otherwise

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Normal Distribution

• The probability density function
• f(x) 1/?(2??) e-(x-?)2/2?2
• generates a normal distribution with mean ? and
  standard deviation ?
• If X is normally distributed with mean ? and
  standard deviation ?, then Y ?X? is normally
  distributed with mean ??? and standard deviation
  ??
• In particular, setting Z (X - ?)/? generates
  the unit normal distribution

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Properties of the Normal Distribution

• The cumulative distribution function
• ?(x) 1/?(2?) ?-?x e-y2/2 dy
• Satisfies the symmetry condition
• ?(-x) 1 - ?(x)
• The distribution function for a normally
  distributed random variable with mean ? and
  standard deviation ? can be written
• Fx(a) ?((a - ?) / ?)
• ?(x) 1 - 1/x?(2?) e-x2/2 for large x

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Normal Distribution Example (Ross)

• An instructor gives an A to those whose test
  score is greater than ??, B for between ? and
  ??, C for ?-? to ?, D for ?-2? to ?-?, and F for
  less than ?-2?
• What is the probability of an A? Of an F?
• Probability of A 1 - ?(1) 0.1587
• Probability of F ?(-2) 0.0288

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Normal and Binomial Distributions

• If Sn denotes the number of successes that occur
  when n independent trials, each resulting in a
  success with probability p, are performed then
  for any a lt b
• P a ? (Sn-np) / ?(np(1-p)) ? b ? ?(b) - ?(a)
• as n ? ?
• Thus we have two good approximations to the
  binomial distribution - the Poisson approximation
  for when np is moderate, and the Normal for when
  np(1-p) is large

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Normal / Binomial Example

• To determine the effectiveness of a certain diet
  in reducing the amount of cholesterol in the
  blood stream, 100 people are put on the diet.
  After they have been on the diet for a sufficient
  time, their cholesterol count will be taken. The
  nutritionist running the experiment has decided
  to endorse the diet if at least 65 of the people
  have a lower cholesterol count after the diet.
  What is the probability that the nutritionist
  endorses the new diet if, in fact, it has no
  effect on the cholesterol level?

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Normal / Binomial Example

• We assume that each persons probability of a
  random reduction is 0.5, independent of any other
  person
• Thus we want to compute the probability that,
  given only random reductions, more than 65
  successes are seen
• PX ? 65
• Transforming as required, this is
• P 3 ? (X-(1000.5) / ?(1000.50.5)
• Which we can approximate as
• 1 - ?(3) 0.0013
• Since n 100 is sufficiently large

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Normal / Binomial Example cont.

• Actually, we should allow for the fact that a
  result of 64.8 will be measured as 65, so what we
  really want is
• PX ? 64.5
• Transforming as required, this is
• P 2.9 ? (X-(1000.5) / ?(1000.50.5)
• Which we can approximate as
• 1 - ?(2.9) 0.0019
• Since n 100 is still sufficiently large

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Exponential Distribution

• The probability density function
• f(x) ?e-?x x ? 0
• 0 x lt 0
• generates an exponential distribution with
  parameter ?
• The cumulative distribution function may be found
  by integration as
• F(a) 1 - e-?x
• The exponential distribution arises frequently as
  the time until some event occurs

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Exponential Distribution Properties

• We define a random variable X to be memoryless if
• P X gt st X gt t P X gt s
• (that is, if you have already waited until t, the
  probability that you will have to wait a further
  s is the same as the initial probability that you
  will have to wait s
• The exponential distribution is the only
  memoryless distribution

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Exponential Distribution Example (Ross)

• Consider a bank with two tellers. Suppose that
  when Mr Smith enters the bank, he discovers that
  Ms Jones is being served by one of the tellers,
  and Mr Brown by the other
• Suppose also that Mr Smith is told that his
  service will begin as soon as either Jones or
  Brown leaves
• If the amount of time spent with a teller is
  exponentially distributed with parameter ?, what
  is the probability that, of the three customers,
  Mr smith is the last to leave?

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Exponential Distribution Example (Ross)

• Clearly, one of Brown and Jones must leave first,
  at which time Smith starts to be served
• At that point, since the exponential distribution
  is memoryless, both Smith and the remaining
  customer have the same distribution of
  probability of length of service
• Thus Smiths probability of being the last to
  leave is 0.5
• Since Jones and Brown have equal distributions
  when Smith arrives, they each have probability
  0.25 of being last to leave

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Gamma Distribution

• The gamma distribution with parameters (t, ?) is
  given by the probability density function
• f(x) ?e-?x(?x)t-1 / ?(t) x ? 0
• 0 x lt 0
• where ? is defined as
• ?(t) ?0? e-yyt-1dy
• The gamma distribution arises frequently as the
  time until n occurrences of some event occur
• The special case in which ? 0.5 and tn/2 is
  known as the ?n2 distribution, and arises
  particularly as the error distribution of an n
  dimensional problem, where the error in each
  coordinate is normally distributed

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Weibull Distribution

• The Weibull distribution with parameters (?, ?,
  ?) is given by the probability density function
• f(x) (?/?)??-1e-?? x gt ?
• 0 x? ?
• where ? is defined as
• ? (x - ?) / ?
• Its cumulative density distribution is
• f(x) 1 - e-?? x gt ?
• 0 x? ?

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Weibull Distribution

• The Weibull distribution is used to approximate
  the failure of items composed of many parts,
  where failure of any of the parts causes the
  whole assembly to fail
• For example, a computer fails when any of its
  components fail
• Hard drive
• Memory
• Cpu
• Bus
• Cache
• Etc.

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Cauchy Distribution

• The Cauchy distribution with parameters ? is
  given by the probability density function
• f(x) 1 / (? 1 (x - ?)2
• It has a similar shape to the normal
  distribution, but with fatter tails
• Values far from the mean are more likely to be
  found than with the normal distribution

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Beta Distribution

• The Beta distribution with parameters (a, b) is
  given by the probability density function
• f(x) 1/B(a,b) xa-1(1-x)b-1 0 lt x 1
• 0 otherwise
• where B is defined as
• B(a,b) ?01 xa-1(1-x)b-1dx

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Beta Distribution

• The Beta distribution gives controllable
  distributions
• representing prior knowledge about the shape of
  solutions
• b controls the symmetry of the distribution
• When b lt a, the distribution is skewed to the
  left
• I.e. small values are more likely
• When b a, the distribution is symmetric
• When b gt a, the distribution is skewed to the
  right
• a controls the general shape of the distribution
  for a b
• When a 1, we get a uniform distribution
• When a lt 1, the distribution is concave upwards
• When a gt1, the distribution is convex upwards

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Functions of a Random Variable

• We often know about the relationship between two
  random variables
• In particular, for two random variables X and Y,
  we may know that Y g(X)
• If we also know the probability density function
  of X, we would like to know the provability
  density function of Y
• To know about the probability density of Y at a
  particular value y, we will have to work from the
  probability density of the x that gives rise to
  y, ie of g-1(y)
• (for this to be well-defined, we will need to
  restrict the form of g)

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Functions of a Random Variable

• Let X be a continuous random variable with
  probability density function fX. Suppose that
  g(x) is a strictly increasing (or strictly
  decreasing) function of x. Then the variable Y
  defined by Y g(X) has a probability density
  function fY given by
• fY(y) fXg-1(y) d/dy g-1(y) when
  g-1(y) is defined 0 when g-1(y) is
  undefined

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Summary

• Continuous Random Variables
• Continuous Distributions
• Uniform
• Normal
• Exponential
• Other Discrete Distributions
• Gamma
• Weibull
• Cauchy
• Beta
• Functions of a Random Variable

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