08: Poisson and More - Stanford University · Catching Pokemon 29 1.Define events/ RVs & state goal...

08: Poisson and MoreLisa Yan and Jerry CainSeptember 30, 2020

1

Lisa Yan and Jerry Cain, CS109, 2020

Quick slide reference

2

3 Poisson 08a_poisson

11 Poisson, continued 08b_poisson_ii

17 Other Discrete RVs 08c_other_discrete

25 Exercises LIVE

33 Poisson approximation LIVE

Poisson RV

3

08a_poisson


Before we start

The natural exponent !:

https://en.wikipedia.org/wiki/E_(mathematical_constant)

4

lim!→#

1 − &'

!= )$%

Jacob Bernoulliwhile studying

compound interest in 1683

https://en.wikipedia.org/wiki/E_(mathematical_constant)


Algorithmic ride sharing

5

!

!

"

""

Probability of " requests from this area in the next 1 minute?On average, ! = 5 requests per minuteSuppose we know:


Algorithmic ride sharing, approximately

At each second:• Independent trial• You get a request (1) or you don’t (0).

Let # = # of requests in minute.% # = & = 5

6

Probability of " requests from this area in the next 1 minute?On average, ! = 5 requests per minute

0 0 1 0 1 … 0 0 0 0 1

1 2 3 4 5 60

# ~ Bin ) = 60, - = 5/60

Break a minute down into 60 seconds:

/ # = " =60"

5

60

!1 −

5

60

"#!

But what if there are two requests in the same second?!



At each millisecond:• Independent trial• You get a request (1) or you don’t (0).


7


Break a minute down into 60,000 milliseconds:

/ # = " =)"

&

)

!1 −

&

)

"#!

…

1 60,000

# ~ Bin ) = 60000, - = &/)

But what if there are two requests in the same millisecond?!



For each time bucket:• Independent trial• You get a request (1) or you don’t (0).


8


Break a minute down into infinitely small buckets:

/ # = " = lim"→%

)"

&

)

!1 −

&

)

"#!

Who wants to see some cool math?

OMG so small

1 ∞

# ~ Bin ), - = &/)


Binomial in the limit

9

/ # = " = lim"→%

)"

&

)

!1 −

&

)

"#!= lim"→$

'!)!(' − ))!

!%'%

1 − l'

"

1 − l'

%

lim!→#

1 − +,

!= .$%

= lim"→$'!

'%(' − ))!!%)!

1 − l'

"

1 − l'

%

Expand

Rearrange

= lim"→$'!

'%(' − ))!!%)!

./0

1 − l'

%Def natural

exponent

= lim"→$' ' − 1 ⋯ ' − ) + 1

'%' − ) !' − ) !

!%)!

./0

1 − l'

%Expand

= lim"→$'%'%

!%)!

./01

Limit analysis

+ cancel= !%)! .

/0Simplify


Algorithmic ride sharing

10

!

!

"

""


* + = , = &&,! )

$% Poisson distribution

Poisson, continued

11

08b_poisson_ii


Consider an experiment that lasts a fixed interval of time.def A Poisson random variable # is the number of successes over the

experiment duration, assuming the time that each success occurs isindependent and the average # of requests over time is constant.

Examples:• # earthquakes per year• # server hits per second• # of emails per day

Poisson Random Variable

12

…

1 End of interval

the time that each success occurs isindependent


Consider an experiment that lasts a fixed interval of time.def A Poisson random variable # is the number of successes over the

experiment duration, assuming the time that each success occurs isindependent and the average # of requests over time is constant.

Examples:• # earthquakes per year• # server hits per second• # of emails per day

Yes, expectation == variance for Poisson RV! More later.

Poisson Random Variable

13

* + = , = )$% &&

,!!~Poi($)Support: {0,1, 2, … }

PMF

% # = &Var # = &Variance

Expectation


Simeon-Denis Poisson

French mathematician (1781 – 1840)• Published his first paper at age 18• Professor at age 21• Published over 300 papers“Life is only good for two things: doing mathematics and teaching it.”

14


EarthquakesThere are an average of 2.79 major earthquakes in the world each year, and major earthquakes occur independently.What is the probability of 3 major earthquakes happening next year?

15

& ' = )!" *#

'!

1. Define RVs

2. Solve

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

1(2

= 3)

Number of earthquakes, 3

,~Poi(*)0 , = *


Are earthquakes really Poissonian?

16

Other Discrete RVs

17

08c_other_discrete


Grid of random variables

18

Number of successes

Ber(/)One trial

Severaltrials

Intervalof time

Bin(', /)

Poi(&) (tomorrow)

One success

Severalsuccesses

Interval of time tofirst success

Time until success

1 = 1


Consider an experiment: independent trials of Ber(-) random variables.def A Geometric random variable # is the # of trials until the first success.

Examples:• Flipping a coin (7 heads = 8) until first heads appears• Generate bits with 7 bit = 1 = 8 until first 1 generated

Geometric RV

19

* + = , = 1 − / &$'/!~Geo(&)Support: {1, 2, … }

PMF

% # =34

Var # =3#44!

Variance

Expectation


Consider an experiment: independent trials of Ber(-) random variables.def A Negative Binomial random variable # is the # of trials until

7 successes.

Examples:• Flipping a coin until 945 heads appears• # of strings to hash into table until bucket 1 has 9 entries

Negative Binomial RV

20

/ # = " =" − 17 − 1

1 − - !#5-5!~NegBin(', &)Support: {9, 9 + 1,… }

PMF

% # =54

Var # =5 3#44!

VarianceExpectation

Geo 8 = NegBin(1, 8)


Grid of random variables

21

Number of successes

Ber(/)One trial

Severaltrials

Intervalof time

Bin(', /)

Poi(&)

Geo(/)

NegBin(2, /)

(tomorrow)

One success

Severalsuccesses

Interval of time tofirst success

Time until success

1 = 1 6 = 1


Catching PokemonWild Pokemon are captured by throwing Pokeballs at them.• Each ball has probability p = 0.1 of capturing the Pokemon.• Each ball is an independent trial.

What is the probability that you catch the Pokemon on the 5th try?

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1. Define events/ RVs & state goal

A. #~Bin 5, 0.1B. #~Poi 0.5C. #~NegBin 5, 0.1D. #~NegBin 1, 0.1E. #~Geo 0.1F. None/other

2. Solve

#~some distribution

Want: 7 : = 5

!


Wild Pokemon are captured by throwing Pokeballs at them.• Each ball has probability p = 0.1 of capturing the Pokemon.• Each ball is an independent trial.


A. #~Bin 5, 0.1B. #~Poi 0.5C. #~NegBin 5, 0.1D. #~NegBin 1, 0.1E. #~Geo 0.1F. None/other

Catching Pokemon

23


2. Solve

#~some distribution

Want: 7 : = 5


2. Solve

Catching PokemonWild Pokemon are captured by throwing Pokeballs at them.• Each ball has probability p = 0.1 of capturing the Pokemon.• Each ball is an independent trial.


24


2. Solve

#~Geo 0.1

Want: 7 : = 5

,~Geo(&) & ' = 1 − & #!$&

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08: Poisson and More - Stanford University · Catching Pokemon 29 1.Define events/ RVs & state goal...

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