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Brian邵彬 · 2024年04月23日

总体是Uniform的话，样本一定是uniform吗

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:

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

$0.086=b^2/12$

$b=1.016$

总体是Uniform的话，样本一定是uniform吗？如果是的话，可以用（0.95-0）/2来计算样本均值吗

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李坏_品职助教 · 2024年04月23日

嗨，爱思考的PZer你好：

总体是Uniform的话，样本一定是服从uniform没错。

但是题目一开始没有告诉我们这个Uniform分布的b（也就是下限）是多少。所以第一小问没法直接用(0+b)/2去计算mean，只能把数字全部加起来除以n得出mean.

到第三问，此时可以用mean的公式，mean = (0+b)/2 = 0.385，0.385是第一小问求出来的mean。

所以得出b = 0.77

----------------------------------------------
虽然现在很辛苦，但努力过的感觉真的很好，加油！

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