概率論考研試題答案

一、單項選擇題(每題2分,共10題)1.設(shè)隨機事件\(A\)與\(B\)互斥,且\(P(A)>0\),\(P(B)>0\),則()A.\(P(A)=1-P(B)\)B.\(P(AB)=P(A)P(B)\)C.\(P(A\cupB)=1\)D.\(P(AB)=0\)【答案:D】2.設(shè)隨機變量\(X\)的概率密度為\(f(x)=\begin{cases}2x,&0<x<1\\0,&其他\end{cases}\),則\(P\{X\leq0.5\}=()A.\(0.25\)B.\(0.5\)C.\(0.75\)D.\(1\)【答案:A】3.設(shè)隨機變量\(X\)服從參數(shù)為\(\lambda\)的泊松分布,且\(P\{X=1\}=P\{X=2\}\),則\(\lambda=()\)A.\(1\)B.\(2\)C.\(3\)D.\(4\)【答案:B】4.設(shè)隨機變量\(X\)和\(Y\)相互獨立,且\(X\simN(1,4)\),\(Y\simN(0,1)\),則\(Z=X-2Y\)服從的分布是()A.\(N(1,8)\)B.\(N(1,6)\)C.\(N(0,8)\)D.\(N(0,6)\)【答案:A】5.設(shè)總體\(X\simN(\mu,\sigma^{2})\),\(X_1,X_2,\cdots,X_n\)是來自總體\(X\)的樣本,\(\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_i\),則\(\frac{\overline{X}-\mu}{\frac{\sigma}{\sqrt{n}}}\)服從的分布是()A.\(N(0,1)\)B.\(t(n-1)\)C.\(\chi^{2}(n-1)\)D.\(F(n-1,n)\)【答案:A】6.設(shè)隨機變量\(X\)的分布函數(shù)為\(F(x)=\begin{cases}0,&x<0\\x^{2},&0\leqx<1\\1,&x\geq1\end{cases}\),則\(P\{0.3<X<0.7\}=()\)A.\(0.2\)B.\(0.4\)C.\(0.6\)D.\(0.8\)【答案:B】7.設(shè)隨機變量\(X\)和\(Y\)的相關(guān)系數(shù)\(\rho_{XY}=0\),則下列結(jié)論正確的是（）A.\(X\)與\(Y\)相互獨立B.\(D(X+Y)=D(X)+D(Y)\)C.\(D(X-Y)=D(X)-D(Y)\)D.\(P\{XY=0\}=1\)【答案:B】8.設(shè)總體\(X\)的概率密度為\(f(x;\theta)=\begin{cases}\frac{1}{\theta},&0<x<\theta\\0,&其他\end{cases}\),\(X_1,X_2,\cdots,X_n\)是來自總體\(X\)的樣本,則\(\theta\)的矩估計量為()A.\(\overline{X}\)B.\(2\overline{X}\)C.\(\frac{1}{n}\sum_{i=1}^{n}X_i^{2}\)D.\(\sqrt{\frac{1}{n}\sum_{i=1}^{n}X_i^{2}}\)【答案:B】9.設(shè)\(A\),\(B\)為隨機事件,\(P(A)=0.6\),\(P(B)=0.4\),\(P(AB)=0.3\),則\(P(A|B)=()\)A.\(0.2\)B.\(0.45\)C.\(0.75\)D.\(0.8\)【答案:C】10.設(shè)隨機變量\(X\)的期望\(E(X)=\mu\),方差\(D(X)=\sigma^{2}\),則對任意正數(shù)\(\varepsilon\),有（）A.\(P\{|X-\mu|\geq\varepsilon\}\leq\frac{\sigma^{2}}{\varepsilon^{2}}\)B.\(P\{|X-\mu|\geq\varepsilon\}\geq\frac{\sigma^{2}}{\varepsilon^{2}}\)C.\(P\{|X-\mu|<\varepsilon\}\leq\frac{\sigma^{2}}{\varepsilon^{2}}\)D.\(P\{|X-\mu|<\varepsilon\}\geq\frac{\sigma^{2}}{\varepsilon^{2}}\)【答案:A】二、多項選擇題(每題2分,共10題)1.設(shè)隨機事件\(A\),\(B\)滿足\(P(A)>0\),\(P(B)>0\),則()A.若\(A\),\(B\)相互獨立,則\(P(A|B)=P(A)\)B.若\(A\),\(B\)互斥,則\(P(A|B)=0\)C.若\(A\),\(B\)對立,則\(P(A|B)=0\)D.若\(A\),\(B\)相互獨立,則\(P(AB)=P(A)P(B)\)【答案:ABD】2.設(shè)隨機變量\(X\)服從正態(tài)分布\(N(\mu,\sigma^{2})\),則(）A.\(E(X)=\mu\)B.\(D(X)=\sigma^{2}\)C.\(P\{X=\mu\}=0\)D.\(X\)的概率密度函數(shù)關(guān)于\(x=\mu\)對稱【答案:ABCD】3.設(shè)隨機變量\(X\)和\(Y\)的聯(lián)合分布函數(shù)為\(F(x,y)\),邊緣分布函數(shù)分別為\(F_X(x)\)和\(F_Y(y)\),則（）A.\(F(x,+\infty)=F_X(x)\)B.\(F(+\infty,y)=F_Y(y)\)C.\(F(+\infty,+\infty)=1\)D.\(F(-\infty,y)=0\)【答案:ABCD】4.設(shè)總體\(X\)服從參數(shù)為\(\lambda\)的泊松分布,\(X_1,X_2,\cdots,X_n\)是來自總體\(X\)的樣本,則（）A.\(\overline{X}\)是\(\lambda\)的無偏估計量B.\(S^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\overline{X})^{2}\)是\(\lambda\)的無偏估計量C.\(\frac{1}{n}\sum_{i=1}^{n}X_i^{2}\)是\(\lambda^{2}+\lambda\)的無偏估計量D.\(\overline{X}\)與\(S^{2}\)相互獨立【答案:ABC】5.設(shè)隨機變量\(X\)和\(Y\)滿足\(D(X+Y)=D(X)+D(Y)\),則（）A.\(X\)與\(Y\)相互獨立B.\(X\)與\(Y\)不相關(guān)C.\(Cov(X,Y)=0\)D.\(\rho_{XY}=0\)【答案:BCD】6.設(shè)隨機事件\(A\),\(B\),\(C\)滿足\(P(ABC)>0\),則()A.\(P(AB|C)=P(A|C)P(B|C)\)（當(dāng)\(A\),\(B\)在\(C\)發(fā)生條件下相互獨立時）B.\(P(A\cupB|C)=P(A|C)+P(B|C)-P(AB|C)\)C.\(P(A|BC)=\frac{P(ABC)}{P(BC)}\)D.\(P(ABC)=P(A)P(B|A)P(C|AB)\)【答案:ABCD】7.設(shè)隨機變量\(X\)的概率密度函數(shù)\(f(x)\)為偶函數(shù),且\(E(X^{2})\)存在,則（）A.\(E(X)=0\)B.\(D(X)=E(X^{2})\)C.\(f(-x)=f(x)\)D.\(P\{X\leq-a\}=P\{X\geqa\}\)【答案:ACD】8.設(shè)總體\(X\)服從均勻分布\(U(a,b)\),\(X_1,X_2,\cdots,X_n\)是來自總體\(X\)的樣本,則（）A.\(\overline{X}\)是\(\frac{a+b}{2}\)的無偏估計量B.\(S^{2}\)是\(\frac{(b-a)^{2}}{12}\)的無偏估計量C.\(a\)的矩估計量為\(\overline{X}-\sqrt{3}S\)D.\(b\)的矩估計量為\(\overline{X}+\sqrt{3}S\)【答案:ABCD】9.設(shè)隨機變量\(X\)和\(Y\)相互獨立,且\(X\simB(n_1,p)\),\(Y\simB(n_2,p)\),則（）A.\(X+Y\simB(n_1+n_2,p)\)B.\(E(X+Y)=(n_1+n_2)p\)C.\(D(X+Y)=(n_1+n_2)p(1-p)\)D.\(Cov(X,Y)=0\)【答案:ABCD】10.設(shè)隨機事件\(A\)與\(B\)滿足\(P(A\cupB)=0.8\),\(P(A)=0.4\),則()A.若\(A\)與\(B\)互斥,則\(P(B)=0.4\)B.若\(A\)與\(B\)相互獨立,則\(P(B)=0.667\)C.\(P(AB)\leq0.4\)D.\(P(\overline{A}\overline{B})=0.2\)【答案:ABCD】三、判斷題(每題2分,共10題)1.若\(P(A)=0\),則\(A\)是不可能事件。(×)2.設(shè)隨機變量\(X\)與\(Y\)相互獨立,則\(D(X-Y)=D(X)-D(Y)\)。（×)3.總體\(X\)的樣本均值\(\overline{X}\)是總體均值\(E(X)\)的無偏估計量。(√)4.若隨機變量\(X\)服從正態(tài)分布\(N(\mu,\sigma^{2})\),則\(P\{X<\mu\}=0.5\)。(√）5.設(shè)\(A\),\(B\)為隨機事件,若\(A\subsetB\),則\(P(A)\leqP(B)\)。（√)6.隨機變量\(X\)的方差\(D(X)\)恒大于\(0\)。(×)7.若隨機變量\(X\)和\(Y\)的相關(guān)系數(shù)\(\rho_{XY}=1\),則\(Y=aX+b\),\(a>0\)。（√)8.總體\(X\)的樣本方差\(S^{2}\)是總體方差\(D(X)\)的無偏估計量。(√)9.設(shè)\(A\),\(B\)為對立事件,則\(P(A)+P(B)=1\)。(√）10.若隨機變量\(X\)的期望\(E(X)\)存在,則\(E(X)\)是一個常數(shù)。（√）四、簡答題(每題5分,共4題)1.簡述概率的公理化定義。答:設(shè)\(E\)是隨機試驗,\(\Omega\)是它的樣本空間,對于\(E\)的每一事件\(A\)賦予一個實數(shù),記為\(P(A)\),若\(P(A)\)滿足:非負(fù)性\(P(A)\geq0\);規(guī)范性\(P(\Omega)=1\);可列可加性,對兩兩互斥事件\(A_1,A_2,\cdots\),有\(zhòng)(P(\bigcup_{i=1}^{\infty}A_i)=\sum_{i=1}^{\infty}P(A_i)\),則稱\(P(A)\)為事件\(A\)的概率。2.簡述隨機變量數(shù)學(xué)期望的定義。答:對于離散型隨機變量\(X\),若其分布律為\(P\{X=x_k\}=p_k\),\(k=1,2,\cdots\),當(dāng)\(\sum_{k=1}^{\infty}|x_k|p_k\)收斂時,\(E(X)=\sum_{k=1}^{\infty}x_kp_k\);對于連續(xù)型隨機變量\(X\),概率密度為\(f(x)\),當(dāng)\(\int_{-\infty}^{+\infty}|x|f(x)dx\)收斂時,\(E(X)=\int_{-\infty}^{+\infty}xf(x)dx\)。3.簡述矩估計法的步驟。答:先求總體的\(k\)階矩\(\mu_k=E(X^k)\)(\(k=1,2,\cdots\)),令樣本\(k\)階矩\(A_k=\frac{1}{n}\sum_{i=1}^{n}X_i^k\)等于總體\(k\)階矩\(\mu_k\),得到關(guān)于未知參數(shù)的方程組,解方程組得到未知參數(shù)的矩估計量。4.簡述正態(tài)分布的性質(zhì)。答:正態(tài)分布概率密度圖像關(guān)于\(x=\mu\)對稱;\(E(X)=\mu\),\(D(X)=\sigma^{2}\);若\(X\simN(\mu,\sigma^{2})\),\(Y=aX+b\)(\(a\neq0\)),則\(Y\simN(a\mu+b,a^{2}\sigma^{