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Gamma, Expoential, Poisson And Chi Squared Distributions



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Gamma, Expoential, Poisson And Chi Squared Distributions

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