If the discrete random variable \(X\) has the funky-ass distribution (FAS), then its range is \(\text{Range}(X)=\mathbb{N}\) and its pmf is

\[ P(X=k)=\binom{k+r-1}{k}(1-p)^kp^r\quad k=0,\,1,\,2,\,3,\,..., \]

where the parameters \(r\in\mathbb{N}\) and \(0<p<1\) are constants. We denote this \(X\sim\text{FAS}(r,\,p)\).

1. Use what you know about probability to find the value of this infinite series:

\[ \sum\limits_{k=0}^\infty \binom{k+r-1}{k}(1-p)^k. \]

2. Compute the MGF of \(X\);
3. Use the definition of the expected value to compute the mean of \(X\);
4. Use the MGF to compute the mean of \(X\) and verify that you get the same answer you got before;
5. Consider an iid collection \(X_i\overset{\text{iid}}{\sim}\text{FAS}(r,\,p)\) and derive the distribution of the sum \(S_n=\sum_{i=1}^nX_i\).