Satellites are launched into space at times distributed according to a Poisson process with rate lambda. Each satellie independently spends a random time (having distribution G) in space before falling to the ground. Find the probability that none of the satellites in the air at time t was launched before time s, where s < t.
Can someone go through this question step by step. Thanks. =]
Restate the question as finding the probability P(s,t) that all satellites launched before time s fall by time t.Originally Posted by feiyingx
Let G(t) be the probability a satellite has life less than t. Then G(t-x) is the probability a satellite launched at time x falls by time t. It is a feature of a Poisson process that the events (here launch times) are uniformly distributed on any bounded subset of the event space. So for one satellite launched at an unknown time during an interval [a,s], say, the probability it falls by time t is
F(a) = (Integral from a to s of G(t-x) dx)/(s-a).
For n sateliites launched at unknown times during [a,s], the probability they all fall by time t is F(a)^n.
It is another feature of a Poisson process with parameter L (short for lambda) is that the number of events occuring on any subset of the event space of size b has a Poisson distribution with parameter Lb. So the probability that the number of satellites launched during [a,s] is n is exp(-L(s-a))(L(s-a))^n/n!. Multiplying by F(a)^n yields the probability that n satellites are launched during [a,s] and they all fall by time t:
exp(-L(s-a))(L(s-a))^n/n! x F(a)^n = exp(-L(s-a))(L(s-a)F(a))^n/n!.
Summing this for n >= 0 yields the probability that all satellites launched during [a,s] fall by time t:
sum from n=0 to infinity of exp(-L(s-a))(L(s-a)F(a))^n/n!
= exp(-L(s-a)) sum from n=0 to infinity of (L(s-a)F(a))^n/n!
= exp(-L(s-a))exp(L(s-a)F(a))
= exp(L(s-a)F(a) - L(s-a))
= exp(L(s-a)(Integral from a to s of G(t-x) dx)/(s-a) - L(s-a))
= exp(L x Integral from a to s of (G(t-x) - 1) dx).
Taking the limit a -> -infinity yields the desired probability that all satellites launched before time s fall by time t:
P(s,t) = exp(L x Integral from -infinity to s of (G(t-x) - 1) dx).
feiyingx, would you please let us know whether this is right or wrong when you get the answer? Thanks.
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