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Gamma, Expoential, Poisson And Chi Squared Distributions
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Gamma, Expoential, Poisson And Chi Squared Distributions

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Gamma, Expoential, Poisson And Chi Squared Distributions

Gamma, Expoential, Poisson And Chi Squared Distributions

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  • 1. 1.9 Gamma, Exponential, Poisson and Chi-Squared Distributions
  • 2. The Gamma Distribution
    The probability density of the gamma distribution is given by
    where () is a value of the gamma function, defined by
    The above improper integral exists (converges) whenever α > 0.
  • 3. The Gamma Distribution
    () = ( -1)(-1) for any  >1.
    Proof:
    () = ( -1)! when  is a positive integer.
    Proof: () = ( -1)(-1)= ( -1) ( -2) (-2)= ….
    = ( -1) ( -2) ….. (1) = ( -1)!
    (1)=1 and (½) =
  • 4. The Gamma Distribution
    f(x)
    1
     = 1,  = 1
     = 1,  = 2
     = 2,  = 3
    1
    2
    3
    4
    5
    6
    7
    0
    Figure: Graph of some gamma probability density functions
  • 5. The Gamma Distribution
    Mean of gamma distribution:
    Proof:
    (put y = x/)
    Use the identity ( + 1) = (), we get
  • 6. The Gamma Distribution
    Variance of gamma distribution:
    Proof:
    Hence
  • 7. MGF of Gamma Distribution –
    MGF of Gamma Distribution
  • 8. Exponential Distribution
    Exponential Distribution: The density function of exponential distribution is given by
    which is the special case of gamma distribution where  = 1.
    Mean and variance of the exponential distribution are given by
  • 9. Memoryless Property of Exponential Distribution:
    Let X be a r.v. that has exponential distribution. Let s,
    t ≥0. Then,
    since the event If X represents lifetime of
    an equipment, then the above equation states that if the
    equipment has been working for time s, then the probability
    that that it will survive an additional time t depends only on t
  • 10. Memoryless Property of Exponential Distribution
    (not on s) and is identical to the probability of survival for time t
    of a new piece of equipment. In that sense, the equipment does
    not remember that it has been in use for time s.
    NOTES:
    (1) The memoryless property simplifies many calculations and is mainly the reason for wide applicability of the exponential model.
    (2) Under this model, an item that has not been failed so far is as good as new.
  • 11. The Gamma Distribution
    Let X be a r.v. that represents the length of the life time of the machine. So, the density function is
    Here, s=7 is its actual life duration to the present time instant. Then,
  • 12. ** Poisson process involves discrete events in a continuous interval of time , length or space.
    (For eg. No. of emission of radioactive gasses from a power plant during 3 months period)
    **
    Avg no. of occurrence of the event per unit of time.
    (egavg no. of customer arriving per hour in booking centre)
    X
    Random variable defined as no. of occurrence of the event to a system in a time interval of size t
    POISSON DISTRIBUTION
  • 13. If X follows poisson distribution then no. of occurrence of the event in the interval of size t has probability density function
    POISSON DISTRIBUTION
  • 14. ** W
    Random variable defined as waiting time between
    successive arrival.( Continuous random variable)
    1) W< t => Waiting time between successive arrival is less
    than t
    2) W>t => Waiting time between successive arrival is
    atleast t
     No arrival during time interval of length t
    POISSON DISTRIBUTION
  • 15. Application of Exponential Distribution
    .
    If in a Poisson process average number of arrivals per unit time is . Let W denote waiting time between successive arrival ( or the time until the first arrival) . W has an exponential distribution with
  • 16. The Gamma Distribution
    Proof :- Let X be a random variable defined
    as no. of arrival to a system in a time
    interval of size t.
    Since X follows poisson distribution then no. of
    arrival in the interval of size t has PDF
  • 17. The Gamma Distribution
    ie P( x arrivals in the time interval t )
    = f( x; λ)
  • 18. The Gamma Distribution
  • 19. The Gamma Distribution
    So if waiting time between successive arrivals be random variable
    with the distribution function
    the probability density of the waiting time between successive
    arrivals given by
    which is an exponential distribution with
  • 20. The Gamma Distribution
    Note: If W represents the waiting time to the first arrival, then
  • 21. The Gamma Distribution
  • 22. This distribution is a special case of Gamma Distribution in which , where r is a positive integer, and . A random variable X of continuous type that has the pdf
    The Chi-square Distribution
  • 23. is said to have a chi-square distribution with parameter r (degree of freedom). The parameters of X are
    The Chi-square Distribution
  • 24. We write that X is to mean that X has a chi-square distribution with r degree of freedom.
    Example – If X has the pdf
    The Chi-square Distribution
    The Chi-square Distribution
  • 25. then X is . Hence, mean is 4 and variance is 8, and the MGF is
    The Chi-square Distribution
  • 26. To compute the probability P(a<X<b), we use the distribution function as
    Tables of the value of distribution function for selected values of a, b and r have been given.
    Example – Let X be . Then
    The Chi-square Distribution