题目

已知二维离散型随机变量(X,Y)的联合概率分布律为X Y -1 1 2-|||--1 0.1 0.2 0.3-|||-2 0.2 0.1 0.1求(1)X和Y的边缘分布率(2)X+Y的分布律(3)D(X)及E(XY)

已知二维离散型随机变量(X,Y)的联合概率分布律为

$\begin{array}{|c|}\hline X\setminus Y &-1 &1 &2\\\hline -1 &0.1 &0.2 &0.3\\\hline 2 &0.2 &0.1 &0.1\\\hline \end{array}$

求(1)X和Y的边缘分布率

(2)X+Y的分布律

(3)D(X)及E(XY)

题目解答

答案

(1) X和Y的边缘分布率

X的边缘分布率是将Y的所有可能值的概率相加，得到

P(X=-1) = 0.1 + 0.2 + 0.3 = 0.6

P(X=2) = 0.2 + 0.1 + 0.1 = 0.4

$\begin{array}{c|cc} X & -1 & 2 \\ \hline P(X) & 0.6 & 0.4 \\ \end{array}$

Y的边缘分布率是将X的所有可能值的概率相加，得到

P(Y=-1) = 0.1 + 0.2 = 0.3

P(Y=1) = 0.2 + 0.1 = 0.3

P(Y=2) = 0.3 + 0.1 = 0.4

$\begin{array}{c|ccc} Y & -1 & 1 & 2 \\ \hline P(Y) & 0.3 & 0.3 & 0.4 \\ \end{array}$

(2) X+Y的分布律

X+Y的可能取值为-2, 0, 1, 3, 4。对应的概率为

P(X+Y=-2) = P(X=-1,Y=-1) = 0.1

P(X+Y=0) = P(X=-1,Y=1) + P(X=2,Y=-1) = 0.2 + 0.2 = 0.4

P(X+Y=1) = P(X=-1,Y=2) + P(X=2,Y=-1) = 0.3 + 0.2 = 0.5

P(X+Y=3) = P(X=2,Y=1) = 0.1

P(X+Y=4) = P(X=2,Y=2) = 0.1

$\begin{array}{c|ccccc} X+Y & -2 & 0 & 1 & 3 & 4 \\ \hline P(X+Y) & 0.1 & 0.4 & 0.5 & 0.1 & 0.1 \\ \end{array}$

(3) D(X)及E(XY)

X的期望E(X)为

$E(X)=-1\times P(X=-1)+2\times P(X=2)=-1\times0.6+2\times0.4=0.2$

X的方差D(X)为

$D(X)=E(X^2)-[E(X)]^2$

其中， $E(X^2)=(-1)^2\times P(X=-1)+2^2\times P(X=2)=1\times0.6+4\times0.4=2.2$ ，所以

$D(X)=2.2-(0.2)^2=2.2-0.04=2.16$

XY的期望E(XY)为

$\begin{array}{l} E(XY) = \sum_{i,j} x_i y_j P(X=x_i,Y=y_j) \\ = -1 \times -1 \times 0.1 + -1 \times 1 \times 0.2 \\ + -1 \times 2 \times 0.3 + 2 \times -1 \times 0.2 \\ + 2 \times 1 \times 0.1 + 2 \times 2 \times 0.1\\ = 0.1 - 0.2 - 0.6 - 0.4 + 0.2 + 0.4 = -0.5 \end{array}$

因此，X的方差D(X)为2.16，XY的期望E(XY)为-0.5。

解析

步骤 1：计算X和Y的边缘分布率
- X的边缘分布率是将Y的所有可能值的概率相加，得到
- P(X=-1) = 0.1 + 0.2 + 0.3 = 0.6
- P(X=2) = 0.2 + 0.1 + 0.1 = 0.4
- Y的边缘分布率是将X的所有可能值的概率相加，得到
- P(Y=-1) = 0.1 + 0.2 = 0.3
- P(Y=1) = 0.2 + 0.1 = 0.3
- P(Y=2) = 0.3 + 0.1 = 0.4

步骤 2：计算X+Y的分布律
- X+Y的可能取值为-2, 0, 1, 3, 4。对应的概率为
- P(X+Y=-2) = P(X=-1,Y=-1) = 0.1
- P(X+Y=0) = P(X=-1,Y=1) + P(X=2,Y=-1) = 0.2 + 0.2 = 0.4
- P(X+Y=1) = P(X=-1,Y=2) + P(X=2,Y=-1) = 0.3 + 0.2 = 0.5
- P(X+Y=3) = P(X=2,Y=1) = 0.1
- P(X+Y=4) = P(X=2,Y=2) = 0.1

步骤 3：计算D(X)及E(XY)
- X的期望E(X)为
- E(X) = -1 * P(X=-1) + 2 * P(X=2) = -1 * 0.6 + 2 * 0.4 = -0.6 + 0.8 = 0.2
- X的方差D(X)为
- D(X) = E(X^2) - [E(X)]^2
- E(X^2) = (-1)^2 * P(X=-1) + 2^2 * P(X=2) = 1 * 0.6 + 4 * 0.4 = 0.6 + 1.6 = 2.2
- D(X) = 2.2 - (0.2)^2 = 2.2 - 0.04 = 2.16
- XY的期望E(XY)为
- E(XY) = (-1)(-1) * P(X=-1,Y=-1) + (-1)(1) * P(X=-1,Y=1) + (-1)(2) * P(X=-1,Y=2) + (2)(-1) * P(X=2,Y=-1) + (2)(1) * P(X=2,Y=1) + (2)(2) * P(X=2,Y=2)
- E(XY) = 1 * 0.1 + (-1) * 0.2 + (-2) * 0.3 + (-2) * 0.2 + 2 * 0.1 + 4 * 0.1
- E(XY) = 0.1 - 0.2 - 0.6 - 0.4 + 0.2 + 0.4 = -0.5