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NewNameStat

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## Homework Statement

## Homework Equations

My professor gives us the following formula: [ tex ] F_j(t) = \int\limits_0^t exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \lambda_j^{\#}(v)dv \Big\} \lambda_j^{\#}(u)du[ / tex ] where [ tex ]\lambda_j^{\#}(t) [ / tex ] are the cause-specific hazard rates for cause $j$.

## The Attempt at a Solution

Assuming we have independent competing risks (we have not yet talked about dependent competing risks so I think it's safe to assume this), $\tilde{F}_j(t) = \int\limits_0^t exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \tilde{\lambda}_j(v)dv \Big\} \tilde{\lambda}_j(u)du$ where $\tilde{\lambda}_j(u)$ is the marginal hazard rate for cause $j$.

So, $\tilde{F}_1(t) = \int\limits_0^t exp\Big\{- \int\limits_0^u 1 dv - \int\limits_0^u .5\sqrt{v} \; dv \Big\} .5\sqrt{u} \; du = .5\int\limits_0^t exp\Big\{- u - \frac{1}{3}u^{3/2}\; dv \Big\} \sqrt{u}\;du$.

However, I do not think this is right because $\tilde{F}_1(\infty)\neq 1$ (it equals something around $0.25$).