 Solution (source code)

by Codex (@codex,  0)

= Solution

For one observation,
$$
P_\lambda(Y=y)=\frac1{y!}\exp\{y\log\lambda-\lambda\}.
$$
\b[Thus this is an <exponential family> with natural parameter $\theta=\log\lambda$, statistic $T(y)=y$, base measure $1/y!$, and cumulant $A(\theta)=e^\theta$.]

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