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

A: No. Every mixed Poisson distribution has a variance greater than or equal to the mean. The compound Poisson distribution 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 $\tau^2<\mu$.

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

A: No. Every mixed Poisson distribution has a variance greater than or equal to the mean. The compound Poisson distribution 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 $\tau^2<\mu$.

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

A: No. Every mixed Poisson distribution has a variance greater than or equal to the mean. The compound Poisson distribution is not so constrained.

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

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

A: No. Every mixed Poisson distribution has a variance greater than or equal to the mean. The compound Poisson distribution 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 $\tau^2<\mu$.

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

A: No. Every mixed Poisson distribution has a variance greater than or equal to the mean. The compound Poisson distribution is not so constrained.

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

A: No. Every mixed Poisson distribution has a variance greater than or equal to the mean. The compound Poisson distribution is not so constrained.

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