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# Essentials of Stochastic Processes (1)

This slide is to learn English and presentation for me.
So, It might have some mistakes. ## Atom

February 26, 2019

## Transcript

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

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

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

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

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

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

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.

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

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

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