How to find the distribution function of compound Poisson process?

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?

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Please see meta.math.stackexchange.com/questions/1803/… for guidelines on asking homework questions. Also, complete sentences, capitalization and punctuation are appreciated, and you can use LaTeX commands to typeset math. – Nate Eldredge May 15 '11 at 21:01

Here is a possible start, using the better notation $X(t) = \sum\nolimits_{i = 1}^{N(t)} {X_i }$. By the law of total probability, conditioning on $N(n)$,

$${\rm P}(X(n) = 20) = \sum\limits_{k = 0}^\infty {{\rm P}(X(n) = 20|N(n) = k){\rm P}(N(n) = k)}.$$ Hence, since the $X_i$ take values in $\{1,2,3,4\}$, $${\rm P}(X(n) = 20) = \sum\limits_{k = 5}^{20} {{\rm P}(X_1 + \cdots + X_k = 20)\frac{{e^{ - n} n^k }}{{k!}}}.$$

Another possibility. Use the probability-generating function of a compound Poisson process (from which you can obtain the probability mass function of $X(n)$ upon differentiation).

EDIT: The probability-generating function of $X(n)$ is given by $$G_n (z):={\rm E}[z^{X(n)} ] = e^{n[P_{X_i } (z) - 1]} ,$$ where $P_{X_i} (z) = \sum\nolimits_{j = 1}^4 {z^j \frac{j}{{10}}}$ is the probability-generating function of $X_i$ (standard exercise). Hence, using that $${\rm P}(X(n) = k) = \frac{{G_n^{(k)} (0)}}{{k!}}$$ (where $G_n^{(k)} (0)$ is the $k$th derivative evaluated at $0$), you can in principle obtain the probability in question, that is ${\rm P}(X(n) = 20)$.

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