91Ó°ÊÓ

We know that the probability density function (pdf) of a distribution function can be calculated by finding its derivative with respect to time \(t\). Let's denote the pdf of \(F\) as \(f(t)\) and the pdf of \(G\) as \(g(t)\). From the exercise, the cumulative distribution function of \(G\) is given by \(\bar{G}(t)=(\bar{F}(t))^{\alpha}\). First, let's find the pdf of \(G\), \(g(t)\). To do that, we need to differentiate \(\bar{G}(t)\) with respect to \(t\): \[ g(t) = \frac{d}{dt} \bar{G}(t) = \frac{d}{dt} (\bar{F}(t))^{\alpha} \] Using the chain rule, we have: \[ g(t) = \alpha (\bar{F}(t))^{\alpha-1} \frac{d}{dt} \bar{F}(t) \] Since \(\bar{F}(t) = 1 - F(t)\) and \(f(t) = \frac{d}{dt} F(t)\), we can rewrite the pdf of G as: \[ g(t) = \alpha (1 - F(t))^{\alpha - 1}(-f(t)) \] Now we have found the pdfs of both \(F\) and \(G\).

The failure rate function for a distribution is defined as the ratio of the pdf to the survival function, which is given by \(1 - \text{cdf}\). Let's denote the failure rate function of \(F\) by \(\lambda_{F}(t)\) and the failure rate function of \(G\) by \(\lambda_{G}(t)\). We can compute the failure rate functions for both distributions: \[ \lambda_{F}(t) = \frac{f(t)}{1 - F(t)} \] \[ \lambda_{G}(t) = \frac{g(t)}{1 - G(t)} = \frac{g(t)}{\bar{G}(t)} \] Substitute the pdf of \(G\) and the survival function of \(G\) in the equation of \(\lambda_{G}(t)\): \[ \lambda_{G}(t) = \frac{\alpha (1 - F(t))^{\alpha - 1}(-f(t))}{(\bar{F}(t))^{\alpha}} \]

Now we need to find the relationship between the failure rate functions of \(F\) and \(G\). Divide \(\lambda_{G}(t)\) by \(\lambda_{F}(t)\): \[ \frac{\lambda_{G}(t)}{\lambda_{F}(t)} = \frac{\alpha (1 - F(t))^{\alpha - 1}(-f(t))}{(\bar{F}(t))^{\alpha}} \cdot \frac{1 - F(t)}{f(t)} \] Simplify the equation: \[ \frac{\lambda_{G}(t)}{\lambda_{F}(t)} = \alpha \frac{(1 - F(t))^{\alpha - 1}}{(\bar{F}(t))^{\alpha - 1}} \] So, the relationship between the failure rate functions of \(G\) and \(F\) is given by: \[ \lambda_{G}(t) = \alpha \lambda_{F}(t) \frac{(1 - F(t))^{\alpha - 1}}{(\bar{F}(t))^{\alpha - 1}} \]