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1.9 Gamma, Exponential, Poisson and Chi-Squared Distributions
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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.
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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) =
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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
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The Gamma Distribution Mean of gamma distribution: Proof: (put y = x/) Use the identity ( + 1) = (), we get
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The Gamma Distribution Variance of gamma distribution: Proof: Hence
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MGF of Gamma Distribution – MGF of Gamma Distribution
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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
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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
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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.
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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,
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** 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
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If X follows poisson distribution then no. of occurrence of the event in the interval of size t has probability density function POISSON DISTRIBUTION
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** 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
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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
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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
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The Gamma Distribution ie P( x arrivals in the time interval t ) = f( x; λ)
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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
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The Gamma Distribution Note: If W represents the waiting time to the first arrival, then
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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
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is said to have a chi-square distribution with parameter r (degree of freedom). The parameters of X are The Chi-square Distribution
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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
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then X is . Hence, mean is 4 and variance is 8, and the MGF is The Chi-square Distribution
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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
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