Results 1 to 5 of 5

Math Help - Gamma Distribution

  1. #1
    Newbie
    Joined
    Dec 2008
    Posts
    11

    Gamma Distribution

    At certain times during the year, a bus company runs a special van holding ten passengers from Iowa City to Chicago. After the opening of sales of the tickets, the time (in minutes) between sales of tickets for the trip has a gamma distribution with
    \alpha = 3 and \theta = 2

    A) Assuming independence, record an integral that gives the probability of being sold out within one hour.

    B) Approximate part A using a normal distribution. Is this justified?


    -----------------------------------------------------------------------
    Not really looking for the answer, just need to know where to begin.
    Where would I even start with this?
    would it be an \int_{0}^{60} of a gamma distribution?
    P(0 < Z < 60)? whats the N(?,?)
    or would I use a Chi-square distribution since \theta is 2 R would be 6...... Kinda lost on where to even begin any help would be appreciated.
    Last edited by tjsfury; December 8th 2008 at 06:37 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by tjsfury View Post
    At certain times during the year, a bus company runs a special van holding ten passengers from Iowa City to Chicago. After the opening of sales of the tickets, the time (in minutes) between sales of tickets for the trip has a gamma distribution with
    \alpha = 3 and \theta = 2

    A) Assuming independence, record an integral that gives the probability of being sold out within one hour.

    B) Approximate part A using a normal distribution. Is this justified?


    -----------------------------------------------------------------------
    Not really looking for the answer, just need to know where to begin.
    Where would I even start with this?
    would it be an \int_{0}^{60} of a gamma distribution?
    P(0 < Z < 60)? whats the N(?,?)
    or would I use a Chi-square distribution since \theta is 2 R would be 6...... Kinda lost on where to even begin any help would be appreciated.
    A) Let T_i be the random variable time between sale of ticket i-1 and ticket i.

    Then the pdf of T_i is f(t_i) = \frac{t_i^2 e^{-t_i/2}}{16} for t_i \geq 0 and zero elsewhere.

    You require

    \Pr(T_1 + + T_2 + \, .... \, + T_{10} \leq 60) = \Pr(10T \leq 60)

    since the T_i are i.i.d. random variables

    = \Pr(T \leq 6) = \frac{1}{16} \int_0^6 t^2 \, e^{-t/2} \, dt.


    B) I suggest applying the Central Limit Theorem. n = 10 is too small to justify applying it.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2008
    Posts
    11
    Quote Originally Posted by mr fantastic View Post
    A) Let T_i be the random variable time between sale of ticket i-1 and ticket i.

    Then the pdf of T_i is f(t_i) = \frac{t_i^2 e^{-t_i/2}}{16} for t_i \geq 0 and zero elsewhere.

    You require

    \Pr(T_1 + + T_2 + \, .... \, + T_{10} \leq 60) = \Pr(10T \leq 60)

    since the T_i are i.i.d. random variables

    = \Pr(T \leq 6) = \frac{1}{16} \int_0^6 t^2 \, e^{-t/2} \, dt.


    B) I suggest applying the Central Limit Theorem. n = 10 is too small to justify applying it.
    UM.... Never used T_i yet, the teacher was out for a while and had a sub teach it he never went over it but im sure im supposed to use it thanks again, can you be my teacher? lol

    Im going to look at this tomorrow until I understand it im sure he will ask something like this on the test
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by mr fantastic View Post
    A) Let T_i be the random variable time between sale of ticket i-1 and ticket i.

    Then the pdf of T_i is f(t_i) = \frac{t_i^2 e^{-t_i/2}}{16} for t_i \geq 0 and zero elsewhere.

    You require

    \Pr(T_1 + + T_2 + \, .... \, + T_{10} \leq 60) = \Pr(10T \leq 60) ....Surely you jest, Mr. F.

    since the T_i are i.i.d. random variables

    = \Pr(T \leq 6) = \frac{1}{16} \int_0^6 t^2 \, e^{-t/2} \, dt.


    B) I suggest applying the Central Limit Theorem. n = 10 is too small to justify applying it.
    I think Mr. F slipped at the point noted above; it is not true that if X and Y are i.i.d. random variables then Pr(X + Y < z) = Pr(2 X < z).

    Here is an alternative approach which requires some knowledge of the Gamma distribution; see Gamma distribution - Wikipedia, the free encyclopedia, for example.

    If X_1, X_2, \dots , X_{10} are independent random variables with a Gamma distribution and parameters \alpha \text{ and } \theta, then their sum has a Gamma distribution with parameters 10 \alpha \text{ and } \theta. So the total time to sell 10 tickets has a Gamma distribution with \alpha = 10 \cdot 3 = 30 \text{ and } \theta = 2. Hence the probability of being sold out in 60 minutes is Pr(Y < 60) where Y has the Gamma distribution just described. You can find this probability from a table of math functions or you can use something electronic. I used an Excel spreadsheet function, GAMMADIST, to find the probability is about 0.5243.
    Last edited by awkward; December 9th 2008 at 05:16 PM. Reason: fixed typo
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by awkward View Post
    I think Mr. F slipped at the point noted above; it is not true that if X and Y are i.i.d. random variables then Pr(X + Y < z) = Pr(2 X < z).

    Here is an alternative approach which requires some knowledge of the Gamma distribution; see Gamma distribution - Wikipedia, the free encyclopedia, for example.

    If X_1, X_2, \dots , X_{10} are independent random variables with a Gamma distribution and parameters \alpha \text{ and } \theta, then their sum has a Gamma distribution with parameters 10 \alpha \text{ and } \theta. So the total time to sell 10 tickets has a Gamma distribution with \alpha = 10 \cdot 3 = 30 \text{ and } \theta = 2. Hence the probability of being sold out in 60 minutes is Pr(Y < 60) where Y has the Gamma distribution just described. You can find this probability from a table of math functions or you can use something electronic. I used an Excel spreadsheet function, GAMMADIST, to find the probability is about 0.5243.
    Quite right. I made the same slip I've often corrected in others. I should have read my own reply in this old thread: http://www.mathhelpforum.com/math-he...a-samples.html. Thanks for the save, awkward.
    Last edited by mr fantastic; December 9th 2008 at 07:56 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Gamma distribution
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: November 11th 2010, 04:14 PM
  2. cumulative distribution function using a gamma distribution
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: September 11th 2010, 10:05 AM
  3. Gamma Distribution
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: March 5th 2009, 04:52 AM
  4. Gamma Distribution
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: November 25th 2008, 02:35 AM
  5. Replies: 0
    Last Post: March 30th 2008, 12:44 PM

Search Tags


/mathhelpforum @mathhelpforum