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Poisson process has rate $\lambda$ and $\tau \perp \!\!\! \perp N(t)$. To find distribution i've started with $P(N(t+\tau)-N(t)=k) = P(N(\tau) = k)$. I know that $N(t) \sim Poiss(\lambda t)$, but i don't know what to do next to find distribution.

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Hint: use the formula $$ P(N(\tau) = k) = \int_0^\infty P(N(s)=k|\tau = s) ae^{-as}\, ds.$$

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