### 澳洲代写｜STAT6039｜Principles of Mathematical Statistics数学统计原理 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University ）Principles of Mathematical Statistics数学统计原理澳洲代写代考辅导服务！

A first course in mathematical statistics with emphasis on applications; probability, random variables, moment generating functions and correlation, sampling distributions, estimation of parameters by the methods of moments and maximum likelihood, hypothesis testing, the central limit theorem, and Bayesian statistics.

## Principles of Mathematical Statistics数学统计原理问题集

Posterior inference: suppose you have a $\operatorname{Beta}(4,4)$ prior distribution on the probability $\theta$ that a coin will yield a ‘head’ when spun in a specified manner. The coin is independently spun ten times, and ‘heads’ appear fewer than 3 times. You are not told how many heads were seen, only that the number is less than 3 . Calculate your exact posterior density (up to a proportionality constant) for $\theta$ and sketch it.

Predictive distributions: consider two coins, $C_1$ and $C_2$, with the following characteristics: $\operatorname{Pr}\left(\right.$ heads $\left.\mid C_1\right)=0.6$ and $\operatorname{Pr}\left(\right.$ heads $\left.\mid C_2\right)=0.4$. Choose one of the coins at random and imagine spinning it repeatedly. Given that the first two spins from the chosen coin are tails, what is the expectation of the number of additional spins until a head shows up?

Posterior distribution as a compromise between prior information and data: let $y$ be the number of heads in $n$ spins of a coin, whose probability of heads is $\theta$.
(a) If your prior distribution for $\theta$ is uniform on the range $[0,1]$, derive your prior predictive distribution for $y$,
$$\operatorname{Pr}(y=k)=\int_0^1 \operatorname{Pr}(y=k \mid \theta) d \theta,$$
for each $k=0,1, \ldots, n$.
(b) Suppose you assign a $\operatorname{Beta}(\alpha, \beta)$ prior distribution for $\theta$, and then you observe $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies between your prior mean, $\frac{\alpha}{\alpha+\beta}$, and the observed relative frequency of heads, $\frac{y}{n}$.
(c) Show that, if the prior distribution on $\theta$ is uniform, the posterior variance of $\theta$ is always less than the prior variance.
(d) Give an example of a $\operatorname{Beta}(\alpha, \beta)$ prior distribution and data $y, n$, in which the posterior variance of $\theta$ is higher than the prior variance.

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