Downloaded from https://www.everyspec.com in Eq. B-6c, The s-expected value of n is Pe2T) for a homogeneous or nonhomogeneous Poisson process. The s-expected value of = .(T)(g/T)O'1 (B-1b) -ln(til/7 A(T) (U (B-lie)n in p T}dt$ The uncertainty in [ is estimated by using the linearized expansion of 2n4, differentials, and the usual formula for variance of a linear In xdx 1+ (B-7) function. In X,(t) In & + In + (0- Win t where x U (tITV. Use Ski, to denote the (B-12a) s-expecred value of L and take the s-ex- pected values of Eq$ W-6. d InW() - d n + d nэ .MLJmm(T) (B-MS) + (0 in t)r in (B-12b) BLjs - m(2)(1 C') (B-so) EL 12 CW(T) (B-Sb) d In & + (1 + )d I ] AVar {(n 1(t)} D- [m(T)l' (B-9) *.
AV~r (In 0^ + (I + o)PxVOr0.•1 Eq. B-8 contains the elements of the Fischer Information Matrix. The covariance + 2(1 + O)ACov {In I, in matrix is the inverse of the Fischer Informa- tion matrix. Therefore the asymptotic + (I + In (e/T)] 2 }/n covariance matrix for 2n a and Rn 0 Is (B-ISa) AVar(ln &) [1/m(T)M (1 + C2) (B-l0a) •(") - (B-13b) ACov(ln &, In ,) - -[/m(T)jC (B-10b) where A() is an estimated, asymptotic AVar(In ( ) = [1/m(T)I (B-100) standard deviation for n(t). where AVar (.) and ACov (') stand for the The "point estimate plus or minus 2 asymptotic variance and covariance, respec- standard deviations" is used to estimate the tively. uncertainty in In WO).
The factor of 2 Is arbitrary; conventional wisdom in the USA In practice these are estimated by using the often uses 3 standard deviations, but that Maximum Ukelihood estimates for a, P, and seems excessive here. There Is no "right" ri4 T); namely, &, 0, n. (t 0$In T. t a Ont). answer, nor does the uncertainty have an exact probabilistic interpretation. But it is The eatimates for m(t) and Vt are very useful tool for engineer% and managers, I. Roughly speaking, the interval of uncertainty A otf n(0/T) 5 (8-ha) Is such that no one has the vaguest idea