Let $\{X(t),t\ge 0\}$ be a compound Poisson process with $X(t)=\sum_{i=1}^{N(t)}X_i$.
Suppose that $\lambda=1$ and $P(\{X_i=j\})=j/10$, $j=1,2,3,4$. Calculate $P\{X(n)=20\}$.
Difficulty
The summands $X_i$ take four different values, and the number of ways in which they add up to $20$ is pretty large. For example, one can calculate the probability of $N(n)=5$ and $X_1=X_2=X_3=X_4=X_5=4$, which is one of many scenarios of $X(n)=20$. But the summation appears unmanageable. Is there a better way?