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Essentials of Stochastic Processes (1)

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February 26, 2019
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Essentials of Stochastic Processes (1)

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February 26, 2019
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  1. Essentials of Stochastic Processes (1)
    Prologue : Review of probability
    B3 English Seminar
    2019/2/26
    Nagaoka University of Technology
    Atom Yoshizawa

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  2. References
    Author : Rick Durrett
    Translators : Norio Konno et al.
    “Essentials of Stochastic Process” ,
    Maruzen Publishing Ltd. (2012)
    2

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  3. Contents
    0.1 Probabilities, Independence
    0.2 Random Variables, Distributions
    0.3 Expected Value, Moments
    3

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  4. 0.1 Probability and independence
    If 1
    , ⋯ ,
    are events ,
    ・pairwise independent :

    =
    (
    ) for each ≠
    ・independent : 1
    ∩ ⋯ ∩
    = 1
    ⋯ (
    )
    for any 1 ≤ 1
    ≤ ⋯ ≤

    Incidentally, if is also an event ,
    ・Bayes’ formula :
    = (∩)

    (∩)
    = ()(|)

    ()(|)
    4

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  5. 0.1 Probability and independence
    Ex) Flip three coins.
    Event : the first and second coins are in the same direction
    Event : the second and third coins are in the same direction
    Event : the third and first coins are in the same direction
    = = =
    2
    4
    =
    1
    2
    5

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  6. 0.1 Probability and independence
    The front and back of the coin are denoted by , .
    ∩ = ∩ = ∩ = ,
    ∩ =
    2
    8
    =
    1
    4
    =
    1
    2

    1
    2
    =
    i.e. and are independent. Similarly and are independent,
    and are also independent.
    hence, , and are pairwise independent .
    6

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  7. 0.1 Probability and independence
    However, the three events , and are not independent .
    ∩ ∩ = ,
    ∩ ∩ =
    2
    8
    =
    1
    4

    1
    2
    3
    =
    7
    If events are pairwise independent,
    they are not always independent.

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  8. 0.2 Stochastic variable, distribution
    Various probability distributions
    ・binomial distribution
    ・geometric distribution
    ・Poisson distribution
    ・uniform distribution
    ・exponential distribution
    ・standard normal distribution
    8

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  9. 0.2 Stochastic variable, distribution
    ・binomial distribution

    : number of successes, : probability to succeed

    = =


    (1 − )− for = 0, ⋯ ,
    ・Poisson distribution
    : stochastic variable, : parameter
    = = −


    for = 0,1,2, ⋯
    9

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  10. : discrete distribution
    The expected value of ℎ() is defined by the following equation
    𝐸 = �

    ℎ ( = )
    𝐸𝐸 : the expected value of ℎ =
    : the expected value of ℎ = (-th order moment)
    10
    0.3 Expected value, moment

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