Essentials of Stochastic Processes (1)

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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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