Questions in category: 概率论 (Probability)
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1. [Exer16-3] Exercise 90 of Book {Devore2017B} P.247

Posted by haifeng on 2020-06-22 10:37:25 last update 2020-06-22 10:37:25 | Answers (1) | 收藏


 

  • Show that $\mathrm{Cov}(X,Y+Z)=\mathrm{Cov}(X,Y)+\mathrm{Cov}(X,Z)$.
  • Let $X_1$ and $X_2$ be quantitative and verbal scores on one aptitude exam and let $Y_1$ and $Y_2$ be corresponding scores on another exam. If $\mathrm{Cov}(X_1,Y_1)=5$, $\mathrm{Cov}(X_1,Y_2)=1$, $\mathrm{Cov}(X_2,Y_1)=2$, and $\mathrm{Cov}(X_2,Y_2)=8$, what is the covariance between the two total scores $X_1+X_2$ and $Y_1+Y_2$?
     

 

2. [Exer16-2] Exercise 87 of Book {Devore2017B} P.247

Posted by haifeng on 2020-06-22 10:36:36 last update 2020-06-22 10:36:36 | Answers (1) | 收藏


 

  • Use the general formula for the variance of a linear combination to write an expression for $V(aX+Y)$. Then let $a=\frac{\sigma_Y}{\sigma_X}$ and show that $\rho\geqslant -1$. [{\it Hint:} Variance is always $\geqslant 0$, and $\mathrm{Cov}(X,Y)=\sigma_X\cdot\sigma_Y\cdot\rho$.]
  • By considering $V(aX-Y)$, conclude that $\rho\leqslant 1$.
  • Use the fact that $V(W)=0$ only if $W$ is a constant to show that $\rho=1$ only if $Y=aX+b$.
     

 

3. [Exer16-1] Proposition of Book {Devore2017B} P.240

Posted by haifeng on 2020-06-22 08:57:55 last update 2020-07-11 15:28:13 | Answers (2) | 收藏


Definition (随机变量的线性组合)

Given a collection of $n$ random variables $X_1,\ldots,X_n$ and $n$ numerical constants $a_1,\ldots,a_n$, the rv
\[
Y=a_1 X_1+\cdots+a_n X_n=\sum_{i=1}^{n}a_i X_i
\]
is called a linear combination of the $X_i$'s.
 

 

Prove the following proposition.

Prop 1. Let $X_1,X_2,\ldots,X_n$ have mean values $\mu_1,\ldots,\mu_n$, respectively, and variances of $\sigma_1^2,\ldots,\sigma_n^2$, respectively.

(1) Whether or not the $X_i$'s are independent,
  \[
  \begin{split}
  E(a_1 X_1+a_2 X_2+\cdots+a_n X_n)&=a_1 E(X_1)+a_2 E(X_2)+\cdots+a_n E(X_n)\\
  &=a_1\mu_1+a_2\mu_2+\cdots+a_n\mu_n
  \end{split}
  \]

(2) If $X_1,\ldots,X_n$ are independent,
  \[
  \begin{split}
  V(a_1 X_1+a_2 X_2+\cdots+a_n X_n)&=a_1^2 V(X_1)+a_2^2 V(X_2)+\cdots+a_n^2 V(X_n)\\
  &=a_1^2\sigma_1^2+a_2^2\sigma_2^2+\cdots+a_n^2\sigma_n^2
  \end{split}
  \]
  and
  \[
  \sigma_{a_1 X_1+\cdots+a_n X_n}=\sqrt{a_1^2\sigma_1^2+\cdots+a_n^2\sigma_n^2}
  \]

(3) For any $X_1,\ldots,X_n$,
  \[
  V(a_1 X_1+\cdots+a_n X_n)=\sum_{i=1}^{n}\sum_{j=1}^{n}a_i a_j\mathrm{Cov}(X_i,X_j)
  \]

 


Then, we get the following proposition.

Prop 2. Let $X_1,X_2,\ldots,X_n$ be a random sample from a distribution with mean value $\mu$ and standard deviation $\sigma$. Then

  • $E(\bar{X})=\mu_{\bar{X}}=\mu$.
  • $V(\bar{X})=\sigma_{\bar{X}}^2=\frac{\sigma^2}{n}$ and $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$.

Here $\bar{X}=\frac{1}{n}(X_1+X_2+\cdots+X_n)$.


In addition, with $T_o=X_1+\cdots+X_n$ (the sample total), $E(T_o)=n\mu$, $V(T_o)=n\sigma^2$, and $\sigma_{T_o}=\sqrt{n}\sigma$.


 

注: 有时也记 $D(X)=V(X)$.

样本方差 $s^2$ 定义为

\[
s^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2
\]

其开方 $s$ 称为样本均方差或样本标准差.

\[
s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2}
\]

样本 $k$ 阶原点矩 $m_k=\frac{1}{n}\sum_{i=1}^{n}X_i^k$, $k=1,2,\ldots$

样本 $k$ 阶中心矩 $M_k=\frac{1}{n}\sum_{i=1}^{n}(X_i-\bar{X})^k$, $k=1,2,\ldots$


 

Prop. $E(s^2)=D(X)=\sigma^2$.

 

4. 协方差的计算公式

Posted by haifeng on 2020-06-08 10:34:34 last update 2020-06-08 10:36:42 | Answers (1) | 收藏


证明:

\[
\mathrm{Cov}(X,Y)=E(XY)-\mu_X\cdot\mu_Y
\]

 

 

回忆, 协方差的公式为

\[
\mathrm{Cov}(X,Y)=E\bigl[(X-\mu_X)(Y-\mu_Y)\bigr]
\]

5. [Exer15-5] Exercise 36 of Book {Devore2017B} P.221

Posted by haifeng on 2020-06-02 09:16:00 last update 2020-06-02 09:16:00 | Answers (1) | 收藏


Show that if $Y=aX+b$ ($a\neq 0$), then $\rho=\mathrm{Corr}(X,Y)=+1$ or $-1$. Under what conditions will $\rho=+1$?
 

6. [Exer15-4] Exercise 35 of Book {Devore2017B} P.221

Posted by haifeng on 2020-06-02 09:15:26 last update 2020-06-02 09:15:26 | Answers (1) | 收藏


 

  • (a) Use the rules of expected value to show that $\mathrm{Cov}(aX+b,cY+d)=ac\mathrm{Cov}(X,Y)$.
  • (b) Use part (a) along with the rules of variance and standard deviation to show that $\mathrm{Corr}(aX+b,cY+d)=\mathrm{Corr}(X,Y)$ when $a$ and $c$ have the same sign.
  • (c) What happens if $a$ and $c$ have opposite signs?
     

 

7. [Exer15-3] Exercise 33 of Book {Devore2017B} P.221

Posted by haifeng on 2020-06-02 09:14:14 last update 2020-06-02 09:14:32 | Answers (1) | 收藏


Use the result of Exercise 28(Here is Question2505) to show that when $X$ and $Y$ are independent, $\mathrm{Cov}(X,Y)=\mathrm{Corr}(X,Y)=0$.
 

8. [Exer15-2] Exercise 28 of Book {Devore2017B} P.221

Posted by haifeng on 2020-06-02 09:12:19 last update 2020-06-02 09:13:17 | Answers (1) | 收藏


Show that if $X$ and $Y$ are independent rv's, then $E(XY)=E(X)\cdot E(Y)$. Then apply this in Exercise 25(Here is Question2504). [{\it Hint:} Consider the continuous case with $f(x,y)=f_X(x)\cdot f_Y(y)$.]
 

9. [Exer15-1] Exercise 25 of Book {Devore2017B} P.220

Posted by haifeng on 2020-06-02 09:11:51 last update 2020-06-02 09:11:51 | Answers (1) | 收藏


A surveyor wishes to lay out a square region with each side having length $L$. However, because of measurement error, he instead lays out a rectangle in which the north-south sides both have length $X$ and the east-west sides both have length $Y$. Suppose that $X$ and $Y$ are independent and the each is uniformly distributed on the interval $[L-A,L+A]$ (where $0 < A < L$). What is the expected area of the resulting rectangle?
 

10. [Exer14-4] Exercise 7 of Book {Devore2017B} P.212

Posted by haifeng on 2020-05-27 18:47:08 last update 2020-05-27 18:47:08 | Answers (1) | 收藏


The joint probability distribution of the number $X$ of cars and the number $Y$ of buses per signal cycle at a proposed left turn lane is displayed in the accompanying joint probability table.

  $y$
$p(x,y)$ 0 1 2
  0 .025 .015 .010
  1 .050 .030 .020
$x$ 2 .125 .075 .050
  3 .150 .090 .060
  4 .100 .060 .040
  5 .050 .030 .020

 

%%Table in LaTeX

\begin{table}[htbp]
\centering
\begin{tabular}{cc|p{0.5in}p{0.5in}p{0.5in}}
 & & & $y$ & \\
$p(x,y)$ &  & 0 & 1 & 2 \\\hline
\multirow{3}{*}{$x$}& 0 & .025 & .015 & .010\\
~& 1 & .050 & .030 & .020\\
~& 2 & .125 & .075 & .050\\
~& 3 & .150 & .090 & .060\\
~& 4 & .100 & .060 & .040\\
~& 5 & .050 & .030 & .020\\
\hline
\end{tabular}
\end{table}

 

  • (a) What is the probability that there is exactly one car and exactly one bus during a cycle?
  • (b) What is the probability that there is at most one car and at most one bus during a cycle?
  • (c) What is the probability that there is exactly one car during a cycle? Exactly one bus?
  • (d) Suppose the left turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflow during a cycle?
  • (e) Are $X$ and $Y$ independent rv's? Explain.

 

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