# Gamma Distribution

• Dec 8th 2008, 05:41 PM
tjsfury
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
$\displaystyle \alpha$ = 3 and $\displaystyle \theta$ = 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?

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Not really looking for the answer, just need to know where to begin.
would it be an $\displaystyle \int_{0}^{60}$ of a gamma distribution?
P(0 < Z < 60)? whats the N(?,?)
or would I use a Chi-square distribution since $\displaystyle \theta$ is 2 R would be 6...... Kinda lost on where to even begin any help would be appreciated.
• Dec 8th 2008, 07:28 PM
mr fantastic
Quote:

Originally Posted by tjsfury
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
$\displaystyle \alpha$ = 3 and $\displaystyle \theta$ = 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.
would it be an $\displaystyle \int_{0}^{60}$ of a gamma distribution?
P(0 < Z < 60)? whats the N(?,?)
or would I use a Chi-square distribution since $\displaystyle \theta$ is 2 R would be 6...... Kinda lost on where to even begin any help would be appreciated.

A) Let $\displaystyle T_i$ be the random variable time between sale of ticket i-1 and ticket i.

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

You require

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

since the $\displaystyle T_i$ are i.i.d. random variables

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

B) I suggest applying the Central Limit Theorem. n = 10 is too small to justify applying it.
• Dec 8th 2008, 07:39 PM
tjsfury
Quote:

Originally Posted by mr fantastic
A) Let $\displaystyle T_i$ be the random variable time between sale of ticket i-1 and ticket i.

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

You require

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

since the $\displaystyle T_i$ are i.i.d. random variables

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

B) I suggest applying the Central Limit Theorem. n = 10 is too small to justify applying it.

UM.... Never used $\displaystyle T_i$ 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
• Dec 9th 2008, 05:14 PM
awkward
Quote:

Originally Posted by mr fantastic
A) Let $\displaystyle T_i$ be the random variable time between sale of ticket i-1 and ticket i.

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

You require

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

since the $\displaystyle T_i$ are i.i.d. random variables

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

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 $\displaystyle X_1, X_2, \dots , X_{10}$ are independent random variables with a Gamma distribution and parameters $\displaystyle \alpha \text{ and } \theta$, then their sum has a Gamma distribution with parameters $\displaystyle 10 \alpha \text{ and } \theta$. So the total time to sell 10 tickets has a Gamma distribution with $\displaystyle \alpha = 10 \cdot 3 = 30 \text{ and } \theta = 2$. Hence the probability of being sold out in 60 minutes is $\displaystyle Pr(Y < 60)$ 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.
• Dec 9th 2008, 07:11 PM
mr fantastic
Quote:

Originally Posted by awkward
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 $\displaystyle X_1, X_2, \dots , X_{10}$ are independent random variables with a Gamma distribution and parameters $\displaystyle \alpha \text{ and } \theta$, then their sum has a Gamma distribution with parameters $\displaystyle 10 \alpha \text{ and } \theta$. So the total time to sell 10 tickets has a Gamma distribution with $\displaystyle \alpha = 10 \cdot 3 = 30 \text{ and } \theta = 2$. Hence the probability of being sold out in 60 minutes is $\displaystyle Pr(Y < 60)$ 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.

(Doh) 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.