Question

If has a hazard rate function${\lambda }_X\left(t\right)$, compute the hazard rate function of $aX$where $a$is a positive constant.

Short answer

The function's hazard rate is,${\lambda }_{aX}\left(t\right)=\frac{1}{a}{\lambda }_X(t/a)$.

Step-by-step solution

Determine the Random hazard variables.

The hazard rate of a random variable is,

${\lambda }_{aX}\left(t\right)=\frac{f_{aX}\left(t\right)}{1-F_{aX}\left(t\right)}$

We know that,

localid="1649618601408" $$\begin{gathered} F_{aX}\left(t\right)=P(aX\le t) \\ =P(X\le t/a) \\ =F_X(t/a) \end{gathered}$$

Equation of the value.

So we get that,

$$\begin{gathered} f_{ax}\left(t\right)=\frac{d}{dt}{F}_{ax}\left(t\right) \\ =\frac{d}{dt}\left(F_X(t/a)\right) \\ =\frac{1}{a}{f}_X(t/a) \end{gathered}$$

Finally, we have that,

$$\begin{gathered} {\lambda }_{aX}\left(t\right)=\frac{f_{aX}\left(t\right)}{1-F_{aX}\left(t\right)} \\ =\frac{\frac{1}{a}{f}_X(t/a)}{1-F_X(t/a)} \\ =\frac{1}{a}{\lambda }_X(t/a) \end{gathered}$$