1. EXPONENTIAL RANDOM VARIABLES AND POISSON PROCESSES
MIKE BAILEY
MSIM 852
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Exponential Random Variables

2. Exponential Random Variables
DEFINITION
Let X be a random variable with distribution function
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Exponential Random Variables

3. Exponential Random Variables
f(x) is the DENSITY FUNCTION
F(x) is the DISTRIBUTION FUNCTION
Fc(x) is the SURVIVAL FUNCTION
l is the RATE, m is the EXPECTED VALUE
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Exponential Random Variables

4. Exponential Random Variables
DENSITY FUNCTION
area under the curve
is 1.0
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Exponential Random Variables

5. DERIVING EXPECTED VALUE
The Definition
of Expectation
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Exponential Random Variables

6. DERIVING EXPECTED VALUE
f(x) = 0 if
x < 0
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Exponential Random Variables

7. DERIVING EXPECTED VALUE
Integration by
parts
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Exponential Random Variables

8. DERIVING EXPECTED VALUE
Integral of the
density function
integrates to 1
“zero times
infinity” uses
L’Hopital’s
Rule
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Exponential Random Variables

9. DERIVING EXPECTED VALUE
Integration by parts
and Induction
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Exponential Random Variables

10. Exponential Random Variables
VARIANCE DERIVATION
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Exponential Random Variables

11. COEFFICIENT OF VARIATION
c.v. defined as the ratio of the mean to the standard deviation
standard deviation is SQRT(VAR(X))=1/l
c.v. for exponentials is always 1.0
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Exponential Random Variables

12. MIN OF TWO EXPONENTIALS
Let X1 and X2 be two exponential random variables
rates l1 and l2
independent
What’s the probability X1 is smaller than X2?
l1 / (l1+l2)
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Exponential Random Variables

13. Exponential Random Variables
MINIMA
Conditional Probability
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Exponential Random Variables

14. Exponential Random Variables
MINIMUMS
Substitution x for X1
and use of the density
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Exponential Random Variables

15. Exponential Random Variables
MINIMUMS
Reuse of the density
function
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Exponential Random Variables

16. Exponential Random Variables
Gather terms,
trick with l1+l2
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Exponential Random Variables

17. Exponential Random Variables
Integral of the
density function
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Exponential Random Variables

18. Exponential Random Variables
Gather terms,
trick with l1+l2
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Exponential Random Variables

19. DISTRIBUTION OF THE MINIMUM
Let Z = min(X1, X2)
then Z is exponentially distributed with rate l1+l2
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Exponential Random Variables

20. Exponential Random Variables
Similar arguments used to prove...
hence these two events are independent!
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Exponential Random Variables

21. Exponential Random Variables
MEET THE SNAILS!
100cm 1
Snail covers 100cm in time X1
X1~expon(1.0 days)
E[X1] = 1/1
P[X1>1] = e-1=0.37
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Exponential Random Variables

22. Exponential Random Variables
100cm 1 1 1 1 n
Let Zn = winning time in an n-snail race
Zn ~ expon(nl)
E[Zn]=1/nl
lim E[Zn] = 0 as n gets large
Discuss common error of taking expectations too soon.
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Exponential Random Variables

23. Exponential Random Variables
MEMORYLESSNESS
Called X’s EXCESS LIFE after s
EXCESS LIFE ~ expon(l)
Lightbulbs in the kitchen
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Exponential Random Variables

24. STRONG MEMORYLESSNESS
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Exponential Random Variables

25. Exponential Random Variables
POISSON PROCESSES
Our first STOCHASTIC PROCESS
Inter-event times ~expon(l)
“stationary” over time
suitable for customers arriving at a service facility from an infinite population
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Exponential Random Variables

26. COMPOUND POISSON PROCESS
Processes can be super-imposed
New process is a PP
rate = summed lambdas
probability that a given event sourced from PPi = ratio of rates
exponentially
distributed
amount of time
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Exponential Random Variables

27. FILTERED POISSON PROCESS
Filter (miss, ignore, disqualify, withstand) events independently with probability 1-p
Result: PP(pl)
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Exponential Random Variables