若总体Z~N（a，σ²)，且a，σ已知，又Z₁，Z₂，Z₃，Z₄为样本，试求分别服从什么分布

设X~N(u,σ²),μ未知,且σ²已知,X₁,...X_n为取自此总体的一个样本,指出下列各式中哪些是统计量,哪些不是,为什么？ <img src='https://img2.soutiyun.com/ask/2020-09-30/970331519602713.png' />

设从总体X~N(μ,σ²)中抽取容量为18的一个样本,u,σ²未知,求: <img src='https://img2.soutiyun.com/ask/2020-09-30/970332837126071.png' />

设<img src='https://img2.soutiyun.com/ask/2020-10-04/970679542502247.png' />是取自总体X-N(μ, σ²)的一个样本，均值μ未知，方差σ²已知.;为使μ的双侧1-a置信区间长度不超过I,则至少需要多大的样本量才能达到？

设<img src='https://img2.soutiyun.com/ask/2020-08-12/966096879233995.png' />,且n≥2为正整数，求Aⁿ-2A^n-1

设X₁, X₂, ... X₉是取自正态总体X~N(μ, σ²⁾的样本,且<img src='https://img2.soutiyun.com/ask/2020-08-09/965847636141377.png' />。 求证：<img src='https://img2.soutiyun.com/ask/2020-08-09/965848226531146.png' />。

已知X₁,X₂,…,X₆是来自正态总体N(0,σ²)的简单随机样本.且 <img src='https://img2.soutiyun.com/ask/2020-08-10/96589894787285.png' /> 求a和n. 解题提示 根据t分布的定义来求.

设<img src='https://img2.soutiyun.com/ask/2020-11-18/974563559946235.png' />为来自总体N(μ,σ2)的简单随机样本,<img src='https://img2.soutiyun.com/ask/2020-11-18/974563569546784.png' />为样本均值,已知<img src='https://img2.soutiyun.com/ask/2020-11-18/974563615737426.png' />是σ²的无偏估计(或ET=σ²),则常数C必为（） A.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563625160965.png' /> B.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563634424495.png' /> C.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563643532016.png' /> D.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563651352464.png' />

设X₁，X₂，...，X_n是取自正态总体N(μ，σ²)的样本，μ与σ均未知，则σ²的矩估计量<img src='https://img2.soutiyun.com/ask/2021-01-05/978692195864823.jpg' />为()。 <img src='https://img2.soutiyun.com/ask/2021-01-05/978692212468773.jpg' />

设总体X~N(μ，σ²)，其中σ²已知，若要检验μ，需用统计量<img src='https://img2.soutiyun.com/ask/2020-12-30/978183754089856.jpg' /> (1)若对单边检验，统计假设为H₀：μ=μ₀(μ₀已知)，H₁：μ＞μ₀，则拒绝区间为(); (2)若单边假设为H₀：μ=μ₀，H₁：μ＜μ₀，则拒绝区间为()。(给定显著性水平为α，样本均值为<img src='https://img2.soutiyun.com/ask/2020-12-30/978183901459285.jpg' />，样本容量为n，且可记u_1-α为标准正态分布的(1-α)分位数。)

设从两个正总体X~N(μ₁，σ₁²)与Y~N(μ₂，σ₂²)中分别抽取容量n₁=16与n₂=10的两个相互独立的样本，计算得其样本函数值 <img src='https://img2.soutiyun.com/ask/2021-01-04/978616694515465.jpg' /> 求置信水平为95%的方差比σ₁²/σ₂²的置信区间。

设随机变量X与Y相互独立且分别服从正态分布N(μ,σ²)与N(μ,2σ²),其中σ是未知参数且σ＞0.记Z=X-Y. (I)求Z的概率f(z;σ²) (II)设<img src='https://img2.soutiyun.com/ask/2020-11-18/974564587212992.png' />为来自总体Z的简单随机样本,求σ²的最大似然估计量<img src='https://img2.soutiyun.com/ask/2020-11-18/974564610926348.png' /> (III)证明<img src='https://img2.soutiyun.com/ask/2020-11-18/974564610926348.png' />为σ²的无偏估计量.

设总体X服从正态分布N(μ，σ²)(σ＞0).从该总体中抽取简单随机样本<img src='https://img2.soutiyun.com/ask/2020-11-18/974556174244797.png' />,其样本均值为<img src='https://img2.soutiyun.com/ask/2020-11-18/974556183114305.png' />求统计量<img src='https://img2.soutiyun.com/ask/2020-11-18/974556216981242.png' />的数学期望EY.

设<img src='https://img2.soutiyun.com/ask/2020-12-30/978170171063951.jpg' />是总体N(μ₁，σ₁²)的容量为n₁的样本方差，<img src='https://img2.soutiyun.com/ask/2020-12-30/978170208302081.jpg' />是总体N(μ₂，σ₂²)的容量为n₂的样本方差，且两总体相互独立，其中μ₁，μ₂已知，σ₁，σ₂未知，求σ₁²/σ₂²的置信度为1-α的置信区间。

设<img src='https://img2.soutiyun.com/ask/2020-11-18/974563559946235.png' />是来自总体X~N(μ,σ²)的样本,其中μ已知,σ²＞0为未知参数,样本均值为<img src='https://img2.soutiyun.com/ask/2020-11-18/974563569546784.png' />,则σ²的最大似然估计量为（） A.<img src='https://img2.soutiyun.com/ask/2020-11-18/97456369359988.png' /> B.<img src='https://img2.soutiyun.com/ask/2020-11-18/97456370198636.png' /> C.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563711307893.png' /> D.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563720210402.png' />

设X为随机变量且EX=μ,DX=σ²,a＞0为常数,则由切比雪夫不等式,有<img src='https://img2.soutiyun.com/ask/2020-10-05/970774230821168.jpg' />().

<img src='https://img2.soutiyun.com/ask/2021-01-21/980120611073066.png' />

设<img src='https://img2.soutiyun.com/ask/2020-11-18/974562941547865.png' />,是来自总体N(0,σ²)的简单随机样本,则可以构造未知参数σ²的无偏估计量(或数学期望为σ²的统计量)（） A.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563028033812.png' /> B.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563036905319.png' /> C.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563044501754.png' /> D.<img src='https://img2.soutiyun.com/ask/2020-11-18/974563052164192.png' />

设<img src='https://img2.soutiyun.com/ask/2020-11-18/974555447058205.png' />为来自总体N(μ,σ²)(σ＞0)的简单随机样本;令<img src='https://img2.soutiyun.com/ask/2020-11-18/974555483547292.png' />则（） A.<img src='https://img2.soutiyun.com/ask/2020-11-18/974555514524064.png' /> B.<img src='https://img2.soutiyun.com/ask/2020-11-18/974555523007549.png' /> C.<img src='https://img2.soutiyun.com/ask/2020-11-18/974555531280022.png' /> D.<img src='https://img2.soutiyun.com/ask/2020-11-18/974555539864513.png' />