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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

DistributionGamma FunctionStochastic Process

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

reference
Reference

Wikipedia

recall distribution of a random variable
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
Recall: versus
  • is never-decreasing
  • is rightward continuous
  • ,
  • A slightly modified formula can apply to :
handout problems 6 7
Handout Problems 6 & 7

Technical

  • Problem 6:
    • Gamma function and integration practice
  • Problem 7:
    • important continuous distributions and their relationships
from bernoulli trials to discrete waiting time handout problems 1 4
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 (?)
poisson pwa s distribution
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
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
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
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 (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
  • 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 o n bernoulli trails
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
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
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.

radioactive disintegrations
Radioactive Disintegrations

Rutherford

Chadwick

Geiger

Geiger Counter

explanation
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
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).