08: poisson and more - stanford university · catching pokemon 29 1.define events/ rvs & state...
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08: Poisson and MoreLisa Yan and Jerry CainSeptember 30, 2020
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Lisa Yan and Jerry Cain, CS109, 2020
Quick slide reference
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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
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08a_poisson
Lisa Yan and Jerry Cain, CS109, 2020
Before we start
The natural exponent !:
https://en.wikipedia.org/wiki/E_(mathematical_constant)
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lim!→#
1 − &'
!= )$%
Jacob Bernoulliwhile studying
compound interest in 1683
Lisa Yan and Jerry Cain, CS109, 2020
Algorithmic ride sharing
5
!
!
"
""
Probability of " requests from this area in the next 1 minute?On average, ! = 5 requests per minuteSuppose we know:
Lisa Yan and Jerry Cain, CS109, 2020
Algorithmic ride sharing, approximately
At each second:• Independent trial• You get a request (1) or you don’t (0).
Let # = # of requests in minute.% # = & = 5
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Probability of " requests from this area in the next 1 minute?On average, ! = 5 requests per minute
0 0 1 0 1 … 0 0 0 0 1
1 2 3 4 5 60
# ~ Bin ) = 60, - = 5/60
Break a minute down into 60 seconds:
/ # = " =60"
5
60
!1 −
5
60
"#!
But what if there are two requests in the same second?!
Lisa Yan and Jerry Cain, CS109, 2020
Algorithmic ride sharing, approximately
At each millisecond:• Independent trial• You get a request (1) or you don’t (0).
Let # = # of requests in minute.% # = & = 5
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Probability of " requests from this area in the next 1 minute?On average, ! = 5 requests per minute
Break a minute down into 60,000 milliseconds:
/ # = " =)"
&
)
!1 −
&
)
"#!
…
1 60,000
# ~ Bin ) = 60000, - = &/)
But what if there are two requests in the same millisecond?!
Lisa Yan and Jerry Cain, CS109, 2020
Algorithmic ride sharing, approximately
For each time bucket:• Independent trial• You get a request (1) or you don’t (0).
Let # = # of requests in minute.% # = & = 5
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Probability of " requests from this area in the next 1 minute?On average, ! = 5 requests per minute
Break a minute down into infinitely small buckets:
/ # = " = lim"→%
)"
&
)
!1 −
&
)
"#!
Who wants to see some cool math?
OMG so small
1 ∞
# ~ Bin ), - = &/)
Lisa Yan and Jerry Cain, CS109, 2020
Binomial in the limit
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/ # = " = lim"→%
)"
&
)
!1 −
&
)
"#!= lim"→$
'!)!(' − ))!
!%'%
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
Lisa Yan and Jerry Cain, CS109, 2020
Algorithmic ride sharing
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!
!
"
""
Probability of " requests from this area in the next 1 minute?On average, ! = 5 requests per minute
* + = , = &&,! )
$% Poisson distribution
Poisson, continued
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08b_poisson_ii
Lisa Yan and Jerry Cain, CS109, 2020
Consider an experiment that lasts a fixed interval of time.def A Poisson random variable # is the number of successes over the
experiment duration, assuming the time that each success occurs isindependent and the average # of requests over time is constant.
Examples:• # earthquakes per year• # server hits per second• # of emails per day
Poisson Random Variable
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…
1 End of interval
the time that each success occurs isindependent
Lisa Yan and Jerry Cain, CS109, 2020
Consider an experiment that lasts a fixed interval of time.def A Poisson random variable # is the number of successes over the
experiment duration, assuming the time that each success occurs isindependent and the average # of requests over time is constant.
Examples:• # earthquakes per year• # server hits per second• # of emails per day
Yes, expectation == variance for Poisson RV! More later.
Poisson Random Variable
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* + = , = )$% &&
,!!~Poi($)Support: {0,1, 2, … }
PMF
% # = &Var # = &Variance
Expectation
Lisa Yan and Jerry Cain, CS109, 2020
Simeon-Denis Poisson
French mathematician (1781 – 1840)• Published his first paper at age 18• Professor at age 21• Published over 300 papers“Life is only good for two things: doing mathematics and teaching it.”
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Lisa Yan and Jerry Cain, CS109, 2020
EarthquakesThere are an average of 2.79 major earthquakes in the world each year, and major earthquakes occur independently.What is the probability of 3 major earthquakes happening next year?
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& ' = )!" *#
'!
1. Define RVs
2. Solve
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10
1(2
= 3)
Number of earthquakes, 3
,~Poi(*)0 , = *
Lisa Yan and Jerry Cain, CS109, 2020
Are earthquakes really Poissonian?
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Other Discrete RVs
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08c_other_discrete
Lisa Yan and Jerry Cain, CS109, 2020
Grid of random variables
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Number of successes
Ber(/)One trial
Severaltrials
Intervalof time
Bin(', /)
Poi(&) (tomorrow)
One success
Severalsuccesses
Interval of time tofirst success
Time until success
1 = 1
Lisa Yan and Jerry Cain, CS109, 2020
Consider an experiment: independent trials of Ber(-) random variables.def A Geometric random variable # is the # of trials until the first success.
Examples:• Flipping a coin (7 heads = 8) until first heads appears• Generate bits with 7 bit = 1 = 8 until first 1 generated
Geometric RV
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* + = , = 1 − / &$'/!~Geo(&)Support: {1, 2, … }
PMF
% # =34
Var # =3#44!
Variance
Expectation
Lisa Yan and Jerry Cain, CS109, 2020
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
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/ # = " =" − 17 − 1
1 − - !#5-5!~NegBin(', &)Support: {9, 9 + 1,… }
PMF
% # =54
Var # =5 3#44!
VarianceExpectation
Geo 8 = NegBin(1, 8)
Lisa Yan and Jerry Cain, CS109, 2020
Grid of random variables
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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
Lisa Yan and Jerry Cain, CS109, 2020
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
!
Lisa Yan and Jerry Cain, CS109, 2020
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.
What is the probability that you catch the Pokemon on the 5th try?
A. #~Bin 5, 0.1B. #~Poi 0.5C. #~NegBin 5, 0.1D. #~NegBin 1, 0.1E. #~Geo 0.1F. None/other
Catching Pokemon
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1. Define events/ RVs & state goal
2. Solve
#~some distribution
Want: 7 : = 5
Lisa Yan and Jerry Cain, CS109, 2020
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.
What is the probability that you catch the Pokemon on the 5th try?
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1. Define events/ RVs & state goal
2. Solve
#~Geo 0.1
Want: 7 : = 5
,~Geo(&) & ' = 1 − & #!$&