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To give a concrete example, say the A-train and the B-train arrive on separate tracks and their inter-arrival times are independent of each other, and let [tex]X[/tex] be the continuous rv representing the amount of time until a train arrives. Say the density functions and distribution functions for the two train lines are given by:

[tex]f_a(x) = \frac{1}{4} e^{-\frac{x}{4}} \qquad F_a(x) = 1 - e^{-\frac{x}{4}} \qquad \mbox{ for } x>0 \mbox{, and 0 otherwise}[/tex]

[tex]f_b(x) = \frac{1}{5} e^{-\frac{x}{5}} \qquad F_b(x) = 1 - e^{-\frac{x}{5}} \qquad \mbox{ for } x>0 \mbox{, and 0 otherwise}[/tex]

so the expected time until an A-train arrives [tex]E_a[x] = 4\mbox{ min.}[/tex]

and the expected time until a B-train arrives [tex]E_b[x] = 5\mbox{ min.}[/tex]

and it is easy to determine the probability for an A-train to arrive within any specific amount of time, and similarly for a B-train.

But how can those functions be combined into a single distribution to express the expected time [tex]E_{a|b}[x][/tex] until ANY train arrives, or the probability that ANY train will arrive within, say 2 minutes?