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010.141 Engineering Mathematics II Lecture 3 Distributions

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Zeta (Zipf) 3. A Note. The notion of 'random variables' is a source of major confusion for students. Mainly because, strictly, a random variable isn't a variable ... – PowerPoint PPT presentation

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Title: 010.141 Engineering Mathematics II Lecture 3 Distributions


1
010.141 Engineering Mathematics IILecture
3Distributions Random Variables
  • Bob McKay
  • School of Computer Science and Engineering
  • College of Engineering
  • Seoul National University
  • Partly based on
  • Sheldon Ross A First Course in Probability

2
Outline
  • Random Variables
  • Cumulative Distribution Functions
  • Discrete Distributions
  • Bernoulli Distribution
  • Binomial Distribution
  • Poisson Distribution
  • Other Discrete Distributions
  • Geometric
  • Negative Binomial
  • Hypergeometric
  • Zeta (Zipf)

3
A Note
  • The notion of random variables is a source of
    major confusion for students
  • Mainly because, strictly, a random variable isnt
    a variable
  • That is, the logical notion of a variable cant
    be directly generalised to a random variable
  • In fact, a random variable is a function
  • This leads many textbooks into extremely
    confusing explanations
  • To try to avoid this, we will give more careful
    definitions than in the textbook

4
?-Algebra
  • A ?-algebra over a set ? is a collection ? of
    subsets of ? satisfying
  • If E is in ?, then so is ?\E
  • The union of countably many sets in ? is also in
    ?
  • ? ? ?
  • (which implies that ? ? ?)
  • That is, ? is a non-empty subset of the power set
    P(?), closed under complementation and countable
    union, and containing ?
  • Note that P(?) is itself a ?-algebra, but it is
    not the only one (or even, the most important one)

5
Measure Space
  • A measure space (?,?,?) is a ?-algebra ? over a
    set ?, together with a measure ? ? ? 0, ?
    satisfying
  • ?(?) 0
  • ?(?E?E E) ?E?E ?(E) for any countable set E of
    disjoint sets from ?
  • That is, a measure space is a sigma algebra with
    a measure, which maps ? to zero, and is countably
    additive
  • Note that P(?) itself may not be measurable
  • In general, it wont be if ? is uncountable,
    which is why we have to go to all this trouble.

6
Probability Space
  • A probability space (?, ?,P) is a measure space
    with
  • P(?) 1
  • Note for probabilities, we usually write P
    rather than ?
  • When ? is a finite set, with ? being the power
    set P(?), this gives us our original probability
    axioms

7
Random Variables
  • Given a probability space (?, ?,P), a random
    variable is a measurable function X ? ? S for
    some set S
  • Usually, S is the real numbers
  • We usually write
  • X gt 0 for ? ? ? X(?) gt 0
  • P(X gt 0) for P(X gt 0)
  • P(X x) for P(X x)

8
Random Variable Example (Ross)
  • Three balls are to be randomly selected (without
    replacement) from an urn containing 20 balls
    numbered 1 to 20
  • If we bet that at least one of the drawn balls
    has a number at least 17, what is the probability
    of winning?
  • PX20 19C2 / 20C3 ? 0.15
  • PX19 18C2 / 20C3 ? 0.134
  • PX18 17C2 / 20C3 ? 0.119
  • PX17 16C2 / 20C3 ? 0.105
  • Overall, PX ?17 ? 0.508

9
Cumulative Distribution Functions
  • Given a real-valued random variable X over a
    probability space (?, ?,P), the Cumulative
    Distribution Function (cdf)
  • FX R ? R
  • can be defined as
  • FX(b) P X ? b
  • We usually just write F instead of FX

10
Properties of the cdf
  • a lt b ? FX(a) ? FX(b)
  • limb?? FX(b) 1
  • limb?-? FX(b) 0
  • FX is right-continuous
  • If
  • limm?? bm b
  • bm1 ? bm for all m
  • Then
  • limm?? F(bm) F(b)

11
Discrete Random Variables
  • A discrete random variable is one which can take
    at most countably many values
  • For a discrete random variable, we can define the
    probability mass function
  • px(a) P X a
  • The probability mass function px(a) can be
    positive for only countably many values of a
  • Hence we can enumerate them x1, x2, ...
  • We also have
  • ?i1? px(xi) 1
  • We usually just write p(a) rather than px(a)

12
Bernoulli Random Variables
  • A Bernoulli random variable is one which can take
    only one of two values (say 0 or 1), so we have
  • p(1) P X 1 p
  • p(0) P X 0 1 - p

13
Binomial Random Variables
  • Suppose we conduct n independent trials with a
    Bernoulli random variable
  • The probability mass function of i successes is
    then given by the (n, p) Binomial Distribution
  • p(i) nCi pi (1 - p)n-i

14
Binomial Example (Ross)
  • Suppose an airplane engine will fail, in flight,
    with probability 1-p, independently between
    engines. Suppose that the flight will crash only
    if more than 50 of the engines fail. When should
    you prefer 4 engines to 2?
  • For four engines
  • p4(OK) 4C2p2(1 - p)2 4C3p3(1 - p)
    4C4p4 6p2(1 - p)2 4p3(1 - p) p4
  • For two engines
  • p2(OK) 2C1p (1 - p) 2C2p2 2p(1 - p) p2
  • p4(OK) ? p2(OK) then reduces to
  • p ? 2/3

15
Binomial Random Variable Properties
  • If X is a (n, p) binomial random variable, then
    as k ranges from 0 to n, p(k) first increases
    monotonically, then decreases monotonically
  • The largest value is when k is the largest
    integer less than or equal to p (n 1)

16
Poisson Random Variables
  • The Binomial Distribution isnt always easy to
    work with
  • Either mathematically or practically
  • We may know that a distribution is binomial, but
    not know - or even care about - n
  • Fortunately, for large n, and for values of p
    small enough that np is moderate, it may be
    approximated
  • Set ? np
  • Then p(i) ? e-??i / i!
  • The distribution p(i) e-??i / i! is known as
    the Poisson
  • It is a probability distribution, because
  • ?i0? p(i) e-? ?i0? ?i / i! e-?e? 1

17
Poisson Example (Ross)
  • Suppose we are counting the number of ?-particles
    given off per second from 1 gram of radioactive
    material.
  • We know that, on average, there are 3.2
    ?-particles per second
  • What is the probability, in any given second,
    that there are at most 2 ?-particles?
  • The gram of material contains a huge number of
    particles, of the order of Avogradros number 6
    1023
  • The probabilities of disintegration of the
    particles are independent
  • Hence the mass obeys a binomial distribution,
    which may be accurately approximated by a Poisson
    distribution with ? 3.2
  • PX ? 2 e-3.2 3.2 e-3.2 (3.2)2/2 e-3.2
    ? 0.382

18
Geometric Random Variables
  • Suppose we perform trials until one success is
    achieved
  • If we let X be the number of trials required, it
    has the form
  • P X n (1 - p)n-1p
  • This is known as the geometric distribution

19
Negative Binomial Random Variables
  • In the same way as we generalised the Bernoulli
    distribution to the binomial, we can generalise
    the geometric distribution by performing trials
    until r successes are achieved
  • This is known as the negative binomial
    distribution
  • It has the form
  • P X n n-1Cr-1(1 - p)n-rpr-1

20
Hypergeometric Random Variables
  • We can generalise the geometric distribution in a
    different way, by assuming that the trials are
    not independent
  • Instead, suppose we have to choose a sample of
    size n by random sampling (without replacement)
    from an urn originally containing Np white balls
    and N (1- p) black
  • If we let X be the total number of white balls
    selected, this generates the hypergeometric
    distribution
  • It has the form
  • P X i NpCk N-NpCn-ik / NCn

21
Hypergeometric Application
  • One important application of the hypergeometric
    distribution is in catch-recatch statistics
  • Suppose we want to estimate the number of fish of
    a particular species living in a lake
  • We catch say r 50 fish, then tag and release
    them
  • We assume fish dont learn
  • ie all fish are equally likely to be caught again
  • Now we catch another, say, n 40 fish
  • Assuming there are N fish in the lake, the number
    i which are tagged should follow the
    hypergeometric distribution
  • But thats the wrong way round
  • We know i (say 4), we want to know N
  • From N, we can estimate Pi(N)
  • We assume that the appropriate N is that which
    maximises Pi(N)
  • In this case, we find N 500

22
The Zipf Distribution
  • For many problems, its reasonable to assume that
    the distribution falls off exponentially in a
    parameter k
  • That is, P X k C / k?1
  • For the distribution to be a probability
    distribution, the probabilities must sum to one
  • This implies that
  • C 1 / ?k1? (1/k)?1

23
Zipf Distribution Examples
  • Examples of known occurrence of Zipf
    distributions include
  • Popularity of websites
  • Linkage of networks
  • Wealth of individuals
  • Popularity of names
  • Frequency of words in documents
  • Financial market volatility
  • Phase transitions in physical systems
  • Events in self-organised critical systems

24
Summary
  • ?-algebras and Measures
  • Random Variables
  • Discrete Distributions
  • Bernoulli Distribution
  • Binomial Distribution
  • Poisson Distribution
  • Other Discrete Distributions
  • Geometric
  • Negative Binomial
  • Hypergeometric
  • Zeta (Zipf)

25
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