设X～N(μ，σ²)，σ＞0，二次方程<img src='https://img2.soutiyun.com/latex/latex.action' />无实根的概率为<img src='https://img2.soutiyun.com/latex/latex.action' />，则μ=( )

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' />。

设x₁,x₂…x_n+1是来自N(μ,σ²)的样本,<img src='https://img2.soutiyun.com/ask/2020-09-17/969203538245107.jpg' />,试求常数c,使得<img src='https://img2.soutiyun.com/ask/2020-09-17/969203561081413.jpg' />服从分布,并指出分布的自由度.

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(X₁,X₂,…,X₁₇)是来自正态分布N(μ,σ²)的一个样本,<img src='https://img2.soutiyun.com/ask/2020-09-17/969203692407925.png' />与S²分别是样本均值与样本方差,求k,使得P{<img src='https://img2.soutiyun.com/ask/2020-09-17/969203707825806.png' />＞μ+kS}=0.95.

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X₁，…，X₁₆是来自N(μ，σ²)的样本，经计算<img src='https://img2.soutiyun.com/ask/2020-08-04/96539878216653.png' />试求<img src='https://img2.soutiyun.com/ask/2020-08-04/965398802528695.png' />

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.

设样本X₁，X₂，...，X_n取自正态总体N(μ，σ₀²)(σ₀²已知)，对检验假设H₀：μ=μ₀，H₁：μ＞μ₀的问题，取拒绝域<img src='https://img2.soutiyun.com/ask/2020-12-30/978192934116923.jpg' /> (1)求此检验犯第一类错误的概率为a时，犯第二类错误的概率β，并讨论它们之间的关系； (2)设μ₀=0.5，σ₀²=0.04，α=0.05，n=9，求μ=0.65时不犯第二类错误的概率。

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' />

设总体X服从正态分布N(μ, σ²) (σ＞0),从总体中抽取简单随机样本<img src='https://img2.soutiyun.com/ask/2020-08-09/965846856163765.png' />，其样本均值为<img src='https://img2.soutiyun.com/ask/2020-08-09/965846906898667.png' />求统计量<img src='https://img2.soutiyun.com/ask/2020-08-09/965846894326948.png' /><img src='https://img2.soutiyun.com/ask/2020-08-09/965846932984159.png' />的数学期望。

设<img src='https://img2.soutiyun.com/ask/2020-11-18/974556250959185.png' />是来自总体X~N(μ，σ²)的简单随机样本,记<img src='https://img2.soutiyun.com/ask/2020-11-18/974556263092879.png' /> 求(I)E(Y); (II)D(Y).

设x₁，…，x_n，xn⁺¹是来自N(μ，σ²)的样本，<img src='https://img2.soutiyun.com/ask/2020-08-04/965399598689233.png' /> 试求常数c，使得<img src='https://img2.soutiyun.com/ask/2020-08-04/965399626587828.png' />服从t分布，并指出分布的自由度.

设总体X~N(0,σ²),X₁,X₂,...,X_n是来自总体X的一个样本. <img src='https://img2.soutiyun.com/ask/2020-09-30/970341001201029.png' />

设X₁,X₂,...,X_n是总体N(μ,σ²)的一个样木,求k使<img src='https://img2.soutiyun.com/ask/2020-09-30/97034115297571.png' /><img src='https://img2.soutiyun.com/ask/2020-09-30/970341180409279.png' />σ的无偏估计.

设两个正态分布总体X~N(μ₁,σ²₁),Y~N(μ₂,σ²₂),X₁,X₂,...,X_m与Y₁,...,Y_n是分别来自相互独立的总体X与Y的简单随机样本,S²₁与S²₂分别是其样本方差,已知m=8,S²₁=8.75,n=10,S²₂=2.66,求P{σ²₁＜σ²₂).

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' />().

设总体X~N(μ,σ²)，μ,σ²,未知，X1,...,Xn是X的简单随机样本，则μ的置信水平至少为0.90的置信区间为()。 <img src='https://img2.soutiyun.com/ask/2021-01-07/978868222705221.jpg' />

设X₁，…，X_n为抽自正态总体N(μ，σ²)的简单随机样本，试证 <img src='https://img2.soutiyun.com/ask/2020-08-04/965410591189968.png' /> 为枢轴量，其中k为已知常数，

设X₁,X₂,…,X_n是来自正态总体N(μ,σ²)的简单随机样本,记 <img src='https://img2.soutiyun.com/ask/2020-08-10/965898914993969.png' />i=1,2,...,n.求Y_i服从的分布及相应的概率密度函数. 解题提示 相互独立的正态分布的随机变量的线性组合仍服从正态分布.

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' />