# Question about Poisson convergence to Gaussian distribution in Jaynes' Probability Theory

I have a problem to derive

$$p(s\mid \phi c_1c_2...c_mI_B) \to A \exp\left[\frac{-c(s-\hat{s})^2}{2\hat{s}^2}\right]\tag{6.151} \quad \text{where} \quad \hat{s} = \frac{c}{m\phi}$$

by following what is said in Jaynes' Probability Theory, 6.151

For example, expanding the logarithm of $$p(s\mid\phi c_1...c_mI_B) = \frac{(m\phi)^{c+1}}{c!}s^c \exp({-ms\phi}) \tag{6.147}$$and retaining only through the quadratic terms, we find for the asymptotic formula a Gaussian distribution $(6.151)$.

Though I found how to change a Poisson distribution into a Gaussian distribution, I can’t figure out how the asymptotic formula is derived.

• Jaynes probability theory Nov 4, 2017 at 13:44