Textbook An Introduction to Mathematical Statistics and Its

Textbook: An Introduction to Mathematical Statistics and Its Applications.

Question Number: Section 4.5.8

Let X(1), X(2), and X(3) be three independent negative binomial random bariables with pdf\'s

Pxi(k) = ( K-1 choose 2 ) (4/5)^(3) (1/5)^(k-3), k=3,4,5, to the infinity, for i=1,2,3

Define X=X(1)+X(2)+X(3).

Find P(10<and equal X <and equal 12)?

Hint: Use the moment-generating functions

I have no clue for this question. Can you please help me out solving this problem?

Solution

P(10<=X<=12)=summation over x=10 to 12 [e^tXP_X]=[e^10t(12C2)((4/5)³)((1/5)¹⁰)]+[e^11t(13C2)((4/5)³)((1/5)¹¹)]+[e^12t(14C2)((4/5)³)((1/5)¹²)]=[e^10t(3.46)10^-6]+[e^11t(8.17)10^-7]+[e^12t(1.91)10^-7].