Chapter 4.5, Problem 8Q

Let $X_1,X_2$, and $X_3$ be three independent negative binomial random variables with pdfs

$p_X\left(k\right)=\left(\begin{array}{c}k-1\\ 2\end{array}\right){\left(\frac{4}{5}\right)}^3{\left(\frac{1}{5}\right)}^{k-3},k=3,4,5,\dots$

for $i=1,2,3$. Define $X=X_1+X_2+X_3$. Find $P\left(10\le X\le 12\right)$. (Hint: Use the moment-generating functions of $X_1,X_2$, and $X_3$ to deduce the pdf of $X$.)