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比如世界 · 2020年04月06日

问一道题：NO.PZ2020010304000035

问题如下：

An experiment yields the following data:

It is hypothesized that the data comes from a uniform ditribution, U(0, b).

a. Calculate the sample mean and variance.

b. What are the unbiased estimators of the mean and variance?

c. Calculate the b in U(0, b) using the formula for the mean of a uniform distribution and the value of the unbiased sample mean found in part b.

d. Calculate the b in U(0, b) using the formula for the variance of a uniform distribution and the value of the unbiased sample variance found in part b.

选项：

解释：

a. Use the standard formual to get the sample variance(here, n=15)

$\mu = n^{-1}\sum_{i=1}^{n}X_i =0.39$

$\sigma^2 = n^{-1}\sum_{i=1}^{n}{(X_i-\mu)}^2 =0.08$

b.The sample mean is already unbiased.

For the variance:

$s^2=n\sigma^2/(n-1) =15*0.080/14=0.086$

c.The mean for a U(a,b) distribution is given as:

μ=(a+b)/2

0.385=(0+b)/2

b=0.77

d. The variance for a U(a,b) distribution is given as:

$\sigma^2=(b-a)^2/12$

$0.086=b^2/12$

$b=1.016$

问题：unbiased estimators of the variance，这个指标衡量的是什么，公式是怎么得到的？

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1 个答案

袁园_品职助教 · 2020年04月07日

同学你好！

在 “The Moments” 这一节视频里讲过，公式如图：

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