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For $\Theta \sim \mathrm{Beta}(1,\beta )$, direct integration of its probability density function gives the displayed tail for $0<t<1$. Increasing $\beta$ strictly decreases every such tail probability, and $\Theta$ converges in probability to zero as $\beta \to \infty$. Its mean is $1/(\beta +1)$ and its variance is $\beta /[(\beta +1{)}^2(\beta +2)]$; both decrease for $\beta >1$. For the variance, logarithmic differentiation gives $1/\beta -2/(\beta +1)-1/(\beta +2)<0$ in that range.

After $S$ successes in $N$ conditionally independent Bernoulli trials, Beta-binomial conjugacy gives posterior mean $(S+1)/(N+\beta +1)$. Its derivative with respect to $\beta$ is negative, while the weight $(\beta +1)/(N+\beta +1)$ assigned to the prior mean is increasing for $N>0$.

Using the expected value of a beta distribution, the posterior mean isEquivalently, it is the weighted averageof the prior mean and the maximum-likelihood estimate $n/S$.

The mean of a shape-rate gamma distribution is shape divided by rate. Hence the posterior mean isBoth weights are positive and sum to one. This is a convex combination of the prior mean and the sample mean, provingFor $n\ge 1$ it lies strictly between them if the two means differ, and equals both if they agree. The rate $b$ acts as the prior weight relative to the $n$ observed exposure units.

The prior mean $1/(\beta +1)$ decreases as $\beta$ increases. More strongly, its cumulative distribution function and upper-tail probability areFor every fixed $t$, that upper-tail probability decreases with $\beta$. The statistician increasingly favors smaller values of $\theta$; as $\beta \to \infty$, the prior distribution concentrates near zero. For $\beta >1$, its variance $\beta /[(\beta +1{)}^2(\beta +2)]$ also decreases. This is concentration of a beta prior near zero.

With the data fixed, the derivatives areThus the prior weight increases and the posterior mean decreases. Their limits are $\lambda \to 1$ and ${\tilde{\theta }}_1\to 0$. Both the changing prior mean and the increasing weight contribute to the change in the posterior mean.

The prior mean, posterior mean, and maximum-likelihood estimate areFor $0<S<N$, the last follows by maximizing $S\log \theta +(N-S)\log (1-\theta )$. At $S=0$ or $S=N$, use the usual binomial likelihood convention that admits the endpoints $0$ and $1$ as maximum-likelihood estimates; if the parameter domain is strictly $(0,1)$, these are boundary suprema rather than attained maxima. The posterior mean remains interior in all cases.

TakinggivesSince $N=nm>0$ and $\beta +1>0$, $0<\lambda <1$. Thus the posterior mean is a convex combination of the prior mean and observed binomial proportion, with larger prior weight when $\beta +1$ is large relative to $N$.

The posterior mean is a convex combination of the prior mean and the usual Poisson maximum-likelihood estimate:Both weights are positive and sum to one, so the mean lies between $r/\lambda$ and $\bar{x}$, strictly between them unless they coincide. If all counts are zero, $\bar{x}=0$ is the boundary estimate in the usual nonnegative Poisson parameter space; with a strictly positive parameter restriction it represents the likelihood supremum at that boundary. The displayed posterior-mean relation is valid in either interpretation.

The exponential distribution prior has rate $\lambda =1/3$ and support $\mu \ge 0$, so its prior density is $(1/3)e^{-\mu /3}{1}_{\{\mu \ge 0\}}$. By Bayes' theorem,where $C$ is a normalizing constant independent of $\mu$.

Completing the square gives a useful exact reference for the numerical sampling. PutThen this normal mean posterior with an exponential prior is a normal distribution with location $m$ and variance $s^2$, truncated below at zero. With $\varphi$ the standard normal distribution density and $\Phi$ its cumulative distribution function,The exponential rate is $1/3$, not three; the prior mean is three. The support restriction must be retained in both the formula and the sampler.

For conditionally independent Poisson observations of mean $\theta$ and a gamma distribution prior of shape $r>0$ and rate $\lambda >0$, the likelihood function contributes ${\theta }^{{\sum }_i{x}_i}{e}^{-n\theta }$. Multiplying by the prior density gives a posterior distribution with shape $r+{\sum }_i{x}_i$ and rate $\lambda +n$. The single-observation update is shape $r+X$ and rate $\lambda +1$.

The posterior mean is the convex combinationThus the prior mean $r/\lambda$ and the sample maximum-likelihood estimate $\bar{x}$ bracket it. Integrating the next conditional Poisson distribution against this posterior distribution gives the Poisson-gamma mixture, a negative binomial distribution; its spread includes uncertainty in $\theta$ as well as the conditional count variation.

The prior mean is the expected value of a statistical parameter under its prior distribution, when this expected value exists. The posterior mean instead uses the posterior distribution after conditioning on observations.