I'm having trouble on solving this problem. Given a Poisson process X with parameter lambda and, indipendently, a random variable T with density f(t)=$\theta*e^{(-\theta t)}$, compute the distribution of X(T). I have no idea of how to do it since I don't really understand the meaning of X(T). Thanks in advance
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Have you considered searching our site for the answer? I give a basic definition at https://stats.stackexchange.com/a/215253/919, for instance. I suspect you might be working with an unusual definition of Poisson process, though, because according to standard definitions, $X(t)$ is almost surely zero for any time $t.$ – whuber Oct 07 '23 at 11:15