How can I solve P(X>28.41) using the chi-squared distribution?

[lemonthree]

I'm not quite sure how to solve the question with two approaches.

Let X be the time taken for the cars to arrive.

Given that 1 car passes every 2 mins, theta = 2.
We are interested in the 10th car that passes so alpha = 10.
Thus I know the distribution is Gamma~(theta=2,alpha=10)
I can solve this using the gamma distribution to find P(X>28.41). (Although the summation is from k= 0 to 9 which is pretty tedious)To relate the chi-squared in this case, r = 20 while theta = 2, but I don't see how I can solve P(X>28.41) because it goes back to the gamma distribution.

How do I solve this using the chi-squared distribution?

 

[Valkarie]

An alternate approach to solve this problem is to make use of the Markov property. Since we know that 1 car passes every 2 minutes, we can calculate the expected time for the 10th car to arrive by summing the expected times of each car that passes. That is,Expected Time for 10th Car = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = 110 minsNow, using the Markov property, we can calculate P(X > 28.41) as the probability of the 10th car arriving after 28.41 minutes, which is given by:P(X > 28.41) = 1 - P(X ≤ 28.41) = 1 - (28.41/110) = 0.7398