The distribution of a random variable $X$ is obtained from the binomial distribution $\mathcal{B}(n ; \Pi)$ by conditioning on $X>0$; here $\Pi \in(0,1)$ is an unknown probability parameter and $n$ is known. Show that the distributions of $X$ form an exponential family and identify the natural sufficient statistic $T$, natural parameter $\Phi$, and cumulant function $k(\phi)$. Using general properties of the cumulant function, compute the mean and variance of $X$ when $\Pi=\pi$. Write down an equation for the maximum likelihood estimate $\widehat{\Pi}$of $\Pi$ and explain why, when $\Pi=\pi$, the distribution of $\widehat{\Pi}$is approximately normal $\mathcal{N}(\pi, \pi(1-\pi) / n)$ for large $n$.

Suppose we observe $X=1$. It is suggested that, since the condition $X>0$ is then automatically satisfied, general principles of inference require that the inference to be drawn should be the same as if the distribution of $X$ had been $\mathcal{B}(n ; \Pi)$ and we had observed $X=1$. Comment briefly on this suggestion.