Exponential, Poisson and Gamma Distribution

Archit Vora

All three distribution models different aspect of same process – poisson process.

Poisson Distribution

  • It is used to predict probability of number of events occurring in fixed amount of time
  • Binomial distribution also models similar thing
    • No of heads in n coin flips
    • It has two parameters, n and p. Where p is probability of success.
  • Shortcoming of binomial
    • We want a single number i.e. k events per hour. Binomial has two – n & p
    • More than one event can occur in unit time. 1 like per hour, 1 like per minute.
  • Below formula
    • lambda events occur in unit time
    • Below PDF is probability of k events in unique time
  • Properties
    • Popularly it is used to model rare events so we see small values of lambda often. But that is not restriction
    • Distribution is asymmetric. There is no such thing as # of events < 0
    • As lambda increases, it looks like normal distribution.
  • Poisson Model Assumptions
    • Average rate of events per unit time is constant
    • Events are independent

Exponential Distribution

  • Poisson – prob (k events in unit time)
  • Exponential – Prob (Amount of time between events) = Prob(amount of time until first event)
  • lambda – # no of events in unit time
    • rate
    • same as poisson
  • Derivation from Poisson
    • Exponential CDF can be derived from Poisson PDF
    • Differentiating it gives exponential PDF
  • Memoryless property
    • P(T > (a+b) | P(a) ) = P( T > b )
    • When memory is require we use weibull distribution
      • Older the car, more likely break down
  • When memory is not required
    • Probability that next bus arrives in less than 10 minutes
    • Probability that server will run without restart for 10k hours
    • Time for cook to prepare potato chips (probability not at a point, some range always)
  • Geometric distribution is counter part of exponential in discrete space
    • Corresponding poisson counterpart is binomial distribution
    • No of throws required to observe heads
    • It is also memory less
  • It is monotonically decreasing distribution
  • Expected value = mean = 1 / lambda

Gamma Distribution

  • Exponential – wait time till first. event
  • Gamma – wait time till k events
    • Two params – k and lambda
    • Probability of observing k events in time t
  • Applications
    • You are in a queue for medical checkup. There are 7 people in front of you. Avg time to check one person is 5 minute. (rate = lambada = 1/5 and k = 7)
  • Literature uses different symbols for above parameters
    • alpha, beta
    • theta, k
  • K can be real number in gamma distribution
    • To restrict k to be integer there is Erlang distribution
  • Gamma function – Gamma ( k ) = ( k – 1 ) !

Reference

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