Stat 135 (LEC 001) Computing Assignment 1

2 – 3厘米长，含有1亿个碱基对。在细胞核中

(组蛋白)，形成一串微珠。它是DNA和蛋白质的混合物。

f (r |θ)= 2 r
θ
2

r
2
θ
2

1. 建立一个样本X1的最大似然估计，…Xn来自这个分布。
(将推导内容包含在报告中)
2. 建立了样本X1的线性回归矩估计器的方法，并对其进行了分析。Xn来自这个分布。
(将推导内容包含在报告中)
3.MLE的近似方差和矩量法的方差是什么

2
4. For each of the 3 experiments, plot the log-likelihood functions and find the MLE’s and their
approximate variances. (Include the plots of and numerical results into the report)
5. Find the method of moments estimates and the variances. (Include the numerical results
into the report)
6. For each experiment, make a histogram (with unit area) of the measurements and plot the
fitted densities on top. Do the fits look reasonable? Is there any appreciable difference
between the maximum likelihood fits and the method of moments fits? (Include the plots
and comments into the report)
7. Show that if X follows this distribution with parameter 1, then Y = θX follows this distribution with parameter θ. Thus it is sufficient to figure out how to generate random variables that
are of the distribution aforementioned with a specific parameter θ0. (Include derivation
into the report)
8. Suppose that X, Y are independent standard Gaussian random variables. Show that √
X2 + Y 2
is following aforementioned distribution. And now write a function what can sample from
this distribution with arbitrary θ value. (Hint: Use change of variable: (X, Y ) → (r, α) in
which r =

X2 + Y 2, α = arctan Y
X
) (Include derivation into the report, and fill in
the function ‘distribution sampler’ which takes input of θ parameter value and
the sample size and outputs the sample)

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