Determine P(N1 = k1, N2 = k2, . . . , Nn = kn|NI = k).

Question

Let Ni denote the number of car arrivals in an interval Ii = (ti , ti + ci ]. Suppose we have n such intervals, i = 1, 2, . . . , n, mutually disjoint and assume that Ni ’s are independent with Ni ∼ P o(ciλ), that is, Poisson distribution with parameter ciλ. Denote the union of these intervals by I, and their total length by c = c1 + c2 + · · · + cn. Given ki ≥ 0 and with k = k1 + k2 + · · · + kn, determine P(N1 = k1, N2 = k2, . . . , Nn = kn|NI = k).

Answer 1

Given,k=Σ_(i=1)^n k_i,c=Σ_(i=1)^n c_i

N¬i ~ Poi(c¬_i λ) ,i=1,2,3,…,n. =>Σ_(i=1)^n N_i=N_I ~ Poi(Σ_(i=1)^n c_i λ)≡Poi(cλ) Then P(N=k)=(e^(-cλ)*(cλ)^k)/k! Now, P(N_1=k_1,N_2=k_2,…,N_n=k_n )=Π_(i=1)^n P(N_i=k_i ) ( As all N_i^' s are mutually disjoint and independent) =>P(N_1=k_1,N_2=k_2,…,N_n=k_n | N=k )=P(N_1=k_1 ⋂▒〖N_2=k_2 ⋂▒〖… ⋂▒〖N_n=k_n ⋂▒〖N=k〗〗〗〗)/P(N=k)

P(N_1=k_1 ⋂▒〖N_2=k_2 ⋂▒〖… ⋂▒〖N_n=k_n ⋂▒〖N=k〗〗〗〗)= P(N_1=k_1 ).P(N_2=k_2│N_1=k_1 ).P(N_3=k_3│N_1=k_1∩N_2=k_2 )…P(N_I=k|P(N_1=k_1 ⋂▒〖N_2=k_2 ⋂▒〖… ⋂▒〖N_n=k_n 〗〗〗)

=> P(N_1=k_1 ⋂▒〖N_2=k_2 ⋂▒〖… ⋂▒〖N_n=k_n ⋂▒〖N=k〗〗〗〗)=P(N_1=k_1 ).P(N_2=k_2 )…P(N_n=k_n ).1 If N_i^ 〖= k〗i,i=1,2,…n,then N_I=k=Σ(i=1)^n k_i. So, P(N_I=k|P(N_1=k_1 ⋂▒〖N_2=k_2 ⋂▒〖… ⋂▒〖N_n=k_n 〗〗〗) = 1 Therefore, P(N_1=k_1,N_2=k_2,…,N_n=k_n | N=k )=(Π_(i=1)^n P(N_i=k_i ))/P(N_I=k) =(k!.Π_(i=1)^n c_i^(k_i ))/(Π_(i=1)^n k_i !) = k! .Π_(i=1)^n (ci/c)^(k_i )/(k_i !)