例题：随机变量函数的p.d.f.

例题1

$X$ has p.d.f. $f(x)$, what is p.d.f. of $Y=aX+b (a>0)$?

Solution: 通过c.d.f.作为桥梁来得到随机变量函数的p.d.f.

\[\begin{align} F_Y(y) &\triangleq P(Y \leq y)=P(aX+b\leq y)\\ &=P(X\leq \frac{y-b}{a})=\int_{-\infty}^{\frac{y-b}{a}}f(x)dx\\ &=F_x(\frac{y-b}{a}) \end{align}\]

对 $F_Y(y)$ 进行求导可以得到 $f_Y(y)$ ,

\[\begin{align} f_Y(y)=F'_Y(y) &=\frac{d}{dy}F_x(\frac{y-b}{a}) \\ &=F'(\frac{y-b}{a})\cdot \frac{1}{a} \\ &= \frac{1}{a}f_X(\frac{y-b}{a}) \end{align}\]

例题2

X $\perp\,!\,!\,!\,!\,!\,!\,!!!!\perp$ Y，求 $Z=X+Y$ 的分布？

Solution:

\[\begin{align} F_Z(a)&=P(Z\leq a)=P(X+Y\leq a) \\ &=\int_{\{(x,y):x+y\leq a\}}f_X(x)f_Y(y)dxdy \\ &=\int_{-\infty}^{+\infty}[\int_{-\infty}^{a-x}f_X(x)f_Y(y)dy]dx \\ &=\int_{-\infty}^{+\infty}f_X(x)F_Y(a-x)dx \\ \end{align}\]

于是有

\[F'_Z(a)=\int_{-\infty}^{+\infty}f_X(x)f_Y(a-x)dx \triangleq f_Z(a)\]