Conditioning on

T_{1} = t

P (T_{1} < T_{2}) = \int_{0}^{\infty} P (T_{2} > t ∣ T_{1} = t) f_{1} (t) d t = \int_{0}^{\infty} f_{1} (t) F_{2} (t) d t,

where the notation in the paper uses

F_{2}

ii

Solution

With common fixed censoring time

c

, the ordering is known exactly when the earlier event occurs by

c

; continuity makes ties have probability zero. Thus the informative event is

{min (T_{1}, T_{2}) \leq c}

and

P (T_{1} < T_{2} ∣ informative) = \frac{\int _{0}^{c} f _{1} ( t ) F _{2} ( t ) d t}{1 - F _{1} ( c ) F _{2} ( c )} .

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