例 2.4.1 (离散随机变量的变换后的分布列)

设 $X$ 的分布列为

$$\left(\begin{array}{ccccc}-2& -1& 0& 1& 2\\ 0.15& 0.2& 0.2& 0.2& 0.25\end{array}\right).$$

试求以下随机变量的分布列

• $Y=X^2$,
• $Z=2X-1$,
• $\Psi =\left|X\right|+1$

解

$Y$

对于 $Y=X^2$, $Y$ 的取值为 4, 1 和 0, 而

$$\begin{array}{rl}P(Y=4)& =P\left(\{X=-2\}\cup \{X=2\}\right)\\ & =P(X=-2)+P(X=2)\\ & =0.15+0.25=0.4\text{.}\end{array}$$ $$\begin{array}{rl}P(Y=1)& =P\left(\{X=-1\}\cup \{X=1\}\right)\\ & =P(X=-1)+P(X=1)\\ & =0.2+0.2=0.4\text{.}\end{array}$$ $$P\left(Y=0\right)=P\left(X=0\right)=0.2,$$

所以

$$Y\sim \left(\begin{array}{ccc}4& 1& 0\\ 0.4& 0.4& 0.2\end{array}\right).$$

$Z$

对于 $Z=2X-1$, $Z$ 的取值为 $-5,-3,-1,1$ 和 3, 而

$$\begin{array}{rl}P(Z=-5)& =P(X=-2)=0.15,\\ P(Z=-3)& =P(X=-1)=0.2,\\ P(Z=-1)& =P(X=0)=0.2,\\ P(Z=1)& =P(X=1)=0.2,\\ P(Z=3)& =P(X=2)=\mathrm{0.25.}\end{array}$$

所以

$$Z\sim \left(\begin{array}{ccccc}-5& -3& -1& 1& 3\\ 0.15& 0.2& 0.2& 0.2& 0.25\end{array}\right).$$

$\Psi$

对于 $\Psi =\left|X\right|+1,\Psi$ 的取值为 3, 2, 和 1, 而

$$\begin{array}{rl}P(\Psi =3)& =P\left(\{X=-2\}\cup \{X=2\}\right)\\ & =P(X=-2)+P(X=2)\\ & =0.15+0.25=0.4\text{.}\end{array}$$ $$\begin{array}{rl}P(\Psi =2)& =P\left(\{X=-1\}\cup \{X=1\}\right)\\ & =P(X=-1)+P(X=1)\\ & =0.2+0.2=0.4\text{.}\end{array}$$ $$P\left(\Psi =1\right)=P\left(X=0\right)=0.2,$$

所以

$$\Psi \sim \left(\begin{array}{ccc}3& 2& 1\\ 0.4& 0.4& 0.2\end{array}\right).$$