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

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

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

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 

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 

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 

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. 

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

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 

: 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