 |
22 |
故障率 (3) |
 |
$$
R(t)=\Pr\lbrace t\lt X\rbrace
$$
$$
F(t)=1-R(t)=1-\Pr\lbrace t\lt X\rbrace
$$
$$
F(t)=\Pr\lbrace X\leq t\rbrace
$$
$$
\lambda=\Pr\{X\leq t+\Delta t~|~ t\lt X\}
$$
$$
\lambda=\frac{1}{\Delta t}\cdot\Pr\{X\leq t+\Delta t~|~ t\lt X\}
$$
$$
\Pr\{A~|B\}=\frac{\Pr\{A\cap B\}}{\Pr\{B\}}, \quad \Pr\{B\}\gt 0
$$
$$
\Pr\{A\cup B\}=\Pr\{A\}+\Pr\{B\}-\Pr\{A\cap B\}
$$
$$
\Leftrightarrow \Pr\{A\cap B\}=\Pr\{A\}+\Pr\{B\}-\Pr\{A\cup B\}
$$
$$
\lambda=\frac{1}{\Delta t}\cdot\frac{\Pr\{t\lt X\cap X\leq t+\Delta t\}}{\Pr\{t\lt X\}}
$$
$$
=\frac{1}{\Delta t}\cdot\frac{\Pr\{t\lt X\}+\Pr\{X\leq t+\Delta t\}-\Pr\{t\lt X\cup X\leq t+\Delta t\}}{\Pr\{t\lt X\}}
$$
$$
=\frac{1}{\Delta t}\cdot \frac{R(t)+F(t+\Delta t)-1}{R(t)}=\frac{1}{\Delta t}\cdot\frac{R(t)-R(t+\Delta t)}{R(t)}
$$
$$ -\lambda=\frac{1}{R(t)}\frac{dR(t)}{dt}=\frac{d}{dt}\ln R(t) $$
