The <cumulative incidence function> for cause $k$ is
$$
F_k(t)=\mathbb P(T\leq t,J=k).
$$
Unlike one minus a cause-specific survivor function, it is the absolute probability of observing cause $k$ by time $t$ in the presence of all competing causes.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 / a / iii / Solution

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{wiki}

For event time $T$ and event type $J$, the cumulative incidence function is $F_k(t)=\mathbb P(T\leq t,J=k)$. It is the absolute probability that cause $k$ occurs by time $t$ before any competing cause.


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