Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 6 / iii

Let

π_{A} = P (T_{A} < T_{B})

, and assume

0 < π_{A} < 1

so that both conditional distributions are defined. By independence, the joint density of

T_{A} = u

and

T_{A} < T_{B}

f_{A} (u) F_{B} (u)

, whereas the joint density of

T_{A} = u

and

T_{B} < T_{A}

f_{A} (u) [1 - F_{B} (u)]

. Normalizing gives

f_{T_{A} ∣ T_{A} < T_{B}} (u) = \frac{f _{A} ( u ) F _{B} ( u )}{π _{A}}, f_{T_{A} ∣ T_{B} < T_{A}} (u) = \frac{f _{A} ( u ) [ 1 - F _{B} ( u )]}{1 - π _{A}} .

Both are densities of the latent event time

T_{A}

. The second is not the density of the observed first event

T_{B}

in cases where B wins. Their integrals are 1 by the definitions of

π_{A}

and its complement.

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