Gamma: Find P[15<x<20] for X~GAM(5,4)

Question

I'm having a lot of trouble with this Gamma distribution problem X~GAM(θ,k). My text book is terrible and doesn't have any examples like this: Find P[15 < X <20] for X~GAM(5,4)

Please note my textbook uses X~GAM(θ,k) unlike some others that use alpha and beta for Gamma distributions instead of θ,k.

Could someone please help?

Answer 1

I worked it out. P[15 < X <20] = P[X<20] - P[X less than or equal to 15]

To get P[X less than or equal to 15] it is equal to 1-P[X>15]. Substituting x=15, theta=5 and n-1=3 into the gamma CDF gives 1 - 0.6472 = 0.353.

To get P[X<20] = 1- 0.4335 = 0.5665 by substituting x=20 into the gamma CDF.

So P[15 < X <20] = 0.5665 - 0.353 = 0.214