Distribution Gamma Function Stochastic Process

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Distribution Gamma Function Stochastic Process. Tutorial 4, STAT1301 Fall 2010, 12OCT2010 , MB103@HKU By Joseph Dong. Reference. Wikipedia. Recall: Distribution of a Random Variable.

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## Distribution Gamma Function Stochastic Process

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### DistributionGamma FunctionStochastic Process

Tutorial 4, STAT1301 Fall 2010, 12OCT2010, MB103@HKUBy Joseph Dong

Reference

Wikipedia

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 .
Recall: versus
• is never-decreasing
• is rightward continuous
• ,
• A slightly modified formula can apply to :
Handout Problems 6 & 7

Technical

• Problem 6:
• Gamma function and integration practice
• Problem 7:
• important continuous distributions and their relationships
• 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 (?)
Poisson [pwa’sɔ̃] 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
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.
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 .
Stochastic Process = Set of RVs + Index Set Sample Path of a Stochastic Process

Discrete-time process

Continuous-time process

Bernoulli Trials (Bernoulli Process)
• Bernoulli Trials (with success probability )
• Discrete-time process
• a sequence of independent and identically distributed (iid) Bernoulli Random Variablesfollowing the common distribution .
• Written where are independent and all .
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.
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.
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.
Examples of Poisson Process
• 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.

Rutherford

Geiger

Geiger Counter

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.
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).