a) Suppose that $T$ ~ Exp$(β)$. For any s > 0 and t > 0, find P(T > t + s|T > t).
b) A long-jump athlete, Jay, jumps a distance in metres which is modelled as D = 7.5 + X, where X ∼ Exp(4.6) and X is independent of the distance jumped on any previous attempt. The world long-jump record is 8.95m. Find the probability that, on any given jump, Jay exceeds the world record.
c) Just before his latest jump, Jay’s personal best was b metres. Given that his latest jump exceeded his personal best, find the probability that it exceeded his personal best by at least 0.5m
d) Comment on the appropriateness of the model for the length of each jump that was set out in Part (b).
In terms of my answers I wrote that:
a) P(T > t + s|T > t) = P(T > t + s and T > t)/P(T > t). Therefore T > t is redundant. So
P(T > t + s|T > t) = P(T > s) = 1 - P(T > s) = exp(-βs)
b) I managed to do well also.
c) This question confuses me a bit. But my guess would be that P( D > b + 0.5 | D > b), which we know will be P(D>0.5). Is this ok?
d) Not sure at all how to answer it.
Could you guys please help me with c and d please?
Thank you
I managed to work out the answers in the end. But pleasse let me know if this is correct.
c) P(D > b + 0.5|D > b) = P(X > b − 7.5 + 0.5|X > b − 7.5) = P(X > 0.5)
d) Independence makes less sense e.g. if there is a following wind on one jump then there may be a following wind on the next jump. A probability of about 1 in 1000 of breaking the world record is reasonable. Less reasonable that when he breaks his own record there is a 1/10 chance of breaking it by 50cm