Let $\mathbb P(L=k) = \frac{140}{223(k+2)}, k = 0, 1, ..., 5.$
Credit portfolio consists of 8 independent loans, where $L_i, i = 1, ..., 8$ is the number of “physically” possible defaults of one loan. $L_i$ hasthe same distribution as $L$.
Let $L_{PF}=L_1+...+L_8.$ Find pmf: $G_{L_{PF}}(t)$.
I started from: $G_{L_{PF}}(t) = \mathbb Et^{L_{PF}}=\mathbb Et^{L_1+...+L_8}=\prod_{i=1}^{8} \mathbb Et^{L_i}=\prod_{i=1}^{8} G_{L_i}(t)$
Then
$G_{L_i}(t) = \mathbb Et^{L_i}=\sum_{k=0}^{\infty}t^k \mathbb P(L_i=k)=\frac{140}{223}\sum_{k=0}^{\infty} \frac{t^k}{k+2} $
How to continue? How can I get pmf of $G_{L_{PF}}(t)$?
You can use that $\ln(1-x)=-\sum\limits_{m=1}^{\infty} \frac{x^m}{m} $ (series expansion at $x=0$).
$$-\sum\limits_{k=0}^{\infty} \frac{t^k}{k+2}\stackrel{m=k+2}{=}-\frac1{t^2}\cdot \sum\limits_{m=2}^{\infty} \frac{t^m}{m}=\frac1{t^2}\cdot \left( \ln(1-t)+t \right)$$
Therefore $G_{L_i}(t)=\frac{140}{223}\cdot \sum\limits_{k=0}^{\infty} \frac{t^k}{k+2}=-\frac{140}{223}\cdot\frac{\ln(1-t)+t}{t^2}$
Now it is straightforward to calculate $G_{L_{PF}}(t) $.