= Solution
Multiplying the <Poisson distribution> <likelihood> by the <Jeffreys prior> gives the <posterior density> kernel
$$
\pi(\lambda_i\mid y_i)\propto\lambda_i^{y_i-1/2}e^{-E_i\lambda_i}.
$$
Thus, in shape-rate convention,
$$
\boxed{\lambda_i\mid y_i\sim\operatorname{Gamma}(y_i+1/2,E_i).}
$$
It is a proper <posterior distribution> even when $y_i=0$, provided $E_i>0$. The <posterior mean> and <posterior variance> are $(y_i+1/2)/E_i$ and $(y_i+1/2)/E_i^2$. This is <Poisson-gamma conjugacy with unequal exposures>, with the zero-rate prior interpreted as an improper kernel.
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