I want to prove that the mass function of the negative binomial distribution satisfies that it adds up to $1\;\forall x\in\mathbb R$.
My textbook starts as follows: $\displaystyle\sum_{k=n}^{\infty}\left(\begin{array}{l}k-1 \\ n-1\end{array}\right) p^{n} q^{k-n}=p^{n} \sum_{l=0}^{\infty}\left(\begin{array}{c}n+l-1 \\ l\end{array}\right) q^{l}=p^{n} \sum_{l=0}^{\infty}\left(\begin{array}{c}-n \\ l\end{array}\right)(-q)^{l}$
I don't understand how they substituted $l$ in ${k-1\choose n-1}$. I would have done the following:
$p^{n} \sum_{l=0}^{\infty}\left(\begin{array}{l}n+l-1 \\ k-l-1\end{array}\right) q^{l}$ - but this way I don't get their result.
Could someone help me with this?
(I am new here, so I'm not allowed to post pictures yet. That's why I'm posting links instead of pictures).
Binomial coefficients are symmetric. \begin{eqnarray*} \binom{ k-1}{ n-1}=\binom{ k-1}{ k-n} \end{eqnarray*} Does this help ? Sha, Highlight the above equation & right click, then choose the option show math as & then TeX command ... it will show you the tex commands to write equations.