DistributionGamma FunctionStochastic ProcessTutorial 4, STAT1301 Fall 2010, 12OCT2010, MB103@HKUBy Joseph Dong

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Reference

Wikipedia

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Recall: Distribution of a Random Variable•One way to describe the random behavior of a

random variable is to give its probability distribution, specifying the probability of taking each element in its range (the sample space).

•The representation of a probability distribution comes either in a differential form: the pdf/pmf, or in an integral form: the cdf.

•The cdf is a never-decreasing, right-continuous function from to .

•The pdf/pmf is a non-negative, normalized function from to a subset of .

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Recall: versus

▫ is never-decreasing▫ is rightward continuous▫,

▫A slightly modified formula can apply to :

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Gamma Function

• ,

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Handout Problems 6 & 7

•Problem 6: ▫Gamma function and integration practice

•Problem 7: ▫important continuous distributions and their relationships

Technical

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From Bernoulli Trials to Discrete Waiting Time (Handout Problems 1-4)• A single Bernoulli trial:

▫Tossing a coin▫Only two outcomes and they are complementary to

each other.• Bernoulli trials: we want to count #success, this

gives rise to a Binomial random variable• Bernoulli trials: we want to know how long we

should wait until the first success (Geometric random variable).

• Bernoulli trials: we want to know how long we should wait until the success (Negative Binomial)

• Bernoulli trials: we want to know how long we should wait between two successes (?)

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Poisson [pwa’sɔ̃B] Distribution

•Poisson Approximation to Binomial (PAB)▫Handout Problem 5

•The true utility of Poisson distribution—Poisson process:▫Sort of the limiting case of Bernoulli trials (use PAB to facilitate thinking)

▫“continuous” Bernoulli trials

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Sequence of Random Variables• A sequence of random variables is an ordered

and countable collection of random variables, usually indexed by integers starting from one: , where can be finite or . ▫Shortly written as ▫A sequence of Random Variables is a discrete-

time stochastic process.▫For example, a sequence of Bernoulli trials is a

discrete-time stochastic process called a Bernoulli process.

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Stochastic Process: Discrete-time and Continuous-time

• A stochastic process is (nothing but) an ordered, not necessarily countable, collection of random variables, indexed by an index set .▫Shortly written as ▫Usually bears a physical meaning of Time▫If is a continuous(discrete) set, we call the

indexed r.v.’s a “continuous(discrete)-time process.”

▫In many continuous-time cases, we choose , and in that case, we can write the stochastic process as .

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Stochastic Process = Set of RVs + Index Set Sample Path of a Stochastic Process

Discrete-time process Continuous-time process

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Bernoulli Trials (Bernoulli Process)

•Bernoulli Trials (with success probability )▫Discrete-time process▫a sequence of independent and identically

distributed (iid) Bernoulli Random Variables following the common distribution .

▫Written where are independent and all .

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Poisson Process•Poisson Process (with intensity )

▫Continuous-time process▫Limiting case of Bernoulli Trails when the

index set becomes continuous.▫“Poisson” in the name because the counts

of success on any interval follows , irrespective of the location of the chosen interval on the time axis.

▫Also if two disjoint time intervals and are chosen, then the counts of success on each of them are independent.

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Discrete Distribution Based On Bernoulli Trails•Bernoulli Distribution , one trial•Binomial Distribution , n trials•Poisson Distribution , ly many trials•Geometric Distribution , indefinitely many

but at least one trial•Negative Binomial Distribution ,

indefinitely many but at least r trials.

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Continuous Distribution Based On Poisson Process• Poisson Distribution (discrete) as building block

▫Distribution of counts on any infinitesimal time interval is , where represents the intensity (a differential concept).

▫Additive: , , and independent, then (Proof: use MGF)

• Exponential Distribution as waiting time until first success/arrival/occurrence or inter-arrival time.

• Gamma Distribution as waiting time until success/arrival/occurrence.

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Examples of Poisson Process

•Radioactive disintegrations•Flying-bomb hits on London•Chromosome interchanges in cells•Connection to wrong number•Bacteria and blood counts

Feller: An Introduction to Probability Theory and Its Applications (3e) Vol. 1. §VI.6.

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Radioactive Disintegrations

Geiger Counter

Geiger Rutherford

Chadwick

18Rutherford, Chadwick, and Ellis’ 1920 Experiment

#intervals

(recorded)

#intervals as predicted by

0 57 54.5439956

1 203 210.9397008

2 383 407.8868524

3 525 525.8111954

4 532 508.371522

5 408 393.2082186

6 273 253.4444078

7 139 140.0219269

8 45 67.68889735

9 27 29.08615451

10 16 16.99712879

N 2608

Intensity (7.5s) 3.867331

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Explanation• There are 57 time intervals (7.5 sec each) recorded

zero emission.• There are 203 time intervals (7.5 sec each)

recorded 1 emission.• ……• There are total 2608 time intervals (7.5 sec each)

involved.• On average, each interval recorded 3.87 emissions.• Use 3.87 as the intensity of the Poisson process

that models the counts of emissions on each of the 2608 intervals.

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What’s the waiting time until recording 40 emissions?•Assuming emission mechanism follows a

Poisson process with intensity over every 7.5s interval, then waiting time until recording the emission follows .

•The waiting time of recording the emission follows and its expected value is 40/3.87=154.8 intervals (each of 7.5s long) or 1161 seconds (a bit more than 19 minutes).