Horst, Thanks a lot.
Fan "Horst Kraemer" <[EMAIL PROTECTED]> wrote in message news:[EMAIL PROTECTED] > On Sat, 8 Nov 2003 16:43:30 -0600, "Fan Yang" <[EMAIL PROTECTED]> > wrote: > > > Suppose arrivals come as a rate lambda Poisson process. Let Au be > > the time since the last arrival up to time u. Precisely, if > > 0 = T0 < T1 < T2 < T3 < ... where {Ti : i >= 1} are the arrival times > > and Ti <= u < Ti+1, then Au = u-Ti which is a number in [0, u]. In > > particular, > > if there are no arrivals in [0, u] then we set At = t. > > You mean "...then we set Au = u" > > This somewhat artificial definition can be interpreted more naturally > as: the first arrival takes place at t=0 with probabilty 1, i.e. > N(0)=1. > > > > The notation Au > > refers to > > age, namely the age of the most recent arrival. > > > > Find the probability distribution of Au, by giving the cumulative > > distribution function Gu(t) = P[Au <= t] for all real t. > > > > ********************************************************** > > The following is my solution. > > > > P[A(u)>t] = P[u-A(u)<u-t] = P[T(i) < u-t] > > = P[no arriavals in (u-t, u)] > > = P[N(u)-N(u-t) = 0] > > = exp(-lamba*t) > > > > Then P[A(u)<=t] = 1-exp(-lamba*t) > > > > Is it right? > > Partially ;-) > > F_past[u](t) = 1-exp(-lamba*t) for 0<=t<u > 1 for t>=u > > It is crucial to note that the cdf is *discontinuous* for t=u due to > the fact that we force an arrival t=0 > > It is different from the waiting time for the next event in the future > which has the distribution > > F_future[u](t) = 1-exp(-lambda*t) 0<=t<oo > > A more natural distribution would be the age of the last arrival up to > t=u under the condition that there was at last one arrival in the past > using the standard definition N(0)=0. > > 1 > F[u](t) = ---------------- * (1-exp(-lambda*t) > 1-exp(-lambda*u) > > > which increases continuously from 0 to 1 in [0,1]. > > -- > Horst > . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
