Delta OnTime Performance at HartsfieldJackson Atlanta International June 2003 June 2015 httpwwwtranstatsbtsgovOTDelayotdelaycause1aspdisplaydataamppn1 Data Model Total Operations 2278897 ID: 225095
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Slide1
Binomial and Geometric Distributions
Delta On-Time Performance at Hartsfield-Jackson Atlanta International (June, 2003 - June, 2015)
http://www.transtats.bts.gov/OT_Delay/ot_delaycause1.asp?display=data&pn=1Slide2
Data / Model
Total Operations: 2,278,897
On-Time Operations: 1,824,432
Proportion On-Time: 1824432/2278897 = .8006 (.80)
Will consider random samples of various sizes from this population of operations
Y ≡ # of On-Time operations out of the sample of n
Y ~ Binomial(n ,
p
Y
= 0.80)
X ≡ # of Flights sampled until the first NOT On-Time Arrival is selected
X ~ Geometric(
p
X
= 0.20)Slide3
Binomial Distribution – Probability FunctionSlide4
Geometric Distribution
Used to model the number of Bernoulli trials needed until the first Success occurs (P(
S
)=
p
)First Success on Trial 1
S,
y = 1 p(1)=p First Success on Trial 2 FS, y = 2 p(2)=(1-p)p First Success on Trial k F…FS, y = k p(k)=(1-p)k-1 pSlide5
Binomial Distribution – Expected ValueSlide6
Geometric Distribution – Expected ValueSlide7
Binomial Distribution – Variance and SDSlide8
Geometric Distribution – Variance and SDSlide9
Binomial Distribution for On-Time FlightsSlide10
Binomial Distributions for n=1,2,3,4,10,25
In EXCEL:
Create a column of values 0,1,2,…,n (Say 0 is in cell A2)
In Cell B2, Type:
=BINOM.DIST(A2,n,p,0)
Copy and paste that cell alongside 1 (A3),…,nNote that the 0 at the end gives P(Y = y) = p(y)
If you use 1 instead, you get P(Y ≤ y) = F(y)Slide11
Several Binomial Distributions with p=0.8Slide12Slide13Slide14Slide15Slide16Slide17Slide18
Geometric Distribution ProbabilitiesSlide19
Geometric Distribution
In EXCEL:
Create a column of values 1,2,…,Y* for some large value of Y* (
Say 1
is in cell A2)
In Cell B2, Type: =NEGBINOM.DIST(A2-1,1,p,0)
Copy and paste that cell alongside 1 (A3),…,Y*
Note that the 0 at the end gives P(Y = y) = p(y)
If you use 1 instead, you get P(Y ≤ y) = F(y)Slide20
Geometric Distribution Probabilities and CDFSlide21
Moment-Generating FunctionSlide22
Moment-Generating Function – Binomial DistributionSlide23
Geometric Distribution – MGFSlide24
Probability-Generating FunctionsSlide25
Probability-Generating Functions - BinomialSlide26
Geometric Distribution – PGF