I think I already solve this problem but I would like to know your opinion about the solution.
Let $\{X(t)\}_{t \geq 0}$ be a poisson process with parameter $\lambda$, and T an exponential random variable with parameter $\alpha$ find the distribution of $X(T)$
My solution: First of all I apply Law of total probability over all posible values of T:
$$P(X(T)=x)= \int_{0}^{\infty} P(X(T)=x \mid T=t) f_T(t) dt $$
Where $f_T(t)$ is the density function of $T$
That integral becomes: $$P(X(T)=x)= \int_{0}^{\infty} e^{-\lambda t} \frac{(\lambda t)^x}{x!} \alpha e^{-\alpha t} dt $$ $$P(X(T)=x)= \frac{\alpha \lambda^x}{x!} \int_{0}^{\infty} e^{-(\lambda+ \alpha) t} t^x dt $$
And the integral is a gamma function (doing a change of variable)
Then the result would be:
$$ P(X(T)=x)= \frac{\alpha \lambda^x }{x!} \frac{1}{(\alpha + \lambda)^{x+1}} \Gamma(x+1)$$
What do you think about this solution?