**Q:** *Is every compound Poisson distribution a mixed Poisson distribution?*

**A:** No. Every <A HREF="https://en.wikipedia.org/wiki/Mixed_Poisson_distribution">mixed Poisson distribution</A> has a variance greater than or equal to the mean. The <A HREF="https://en.wikipedia.org/wiki/Compound_Poisson_distribution#Properties">compound Poisson distribution</A> is not so constrained.

If the i.i.d. random variables that are compounded have first moment $\mu$ and second moment $\tau^2\geq\mu^2$, then the compound Poisson distribution has mean $\lambda t\mu$ and variance $\lambda t\tau^2$ (adding $N(t)$  variables with rate $\lambda$). So the variance will be smaller than the mean if $\tau^2<\mu$.