Question

6. Conditioning independent Poisson variables on their sum. Let N, be independent Poisson variables with parameters di. Think of the N, as the number of points of a Poissson scatter in disjoint parts of the plane with areas li, where the mean intensity is one point per unit area. a) What is the conditional joint distribution of (N1, ... , Nm) given Ni+...+Nm = n? (Hint: See Exercise 5 for a special case.)

Answer 1

63 72 -Zdi 1=1 L=1 Given that Ni u Poisson (^i) independently vi therefore N,+Net + Non ~ Poisson (24) (reproductive propeoty) Suppose, To Ni. then PCT=n) 11. Now, P[N=r, Ny=nz, Tan] S P(N= = R1, Ny Raj am-i, Nm = nm) git n-ng P(T= n) q=1 n=0,1,- Nm - Nm Nm- 2 ---- m2 if se te nj II P (Ny = j) =1 P(T= n) Ni's Ni's age independent) 0 -*1.000 ni T e. n! - ()" e j=1 Rm 12 in am Ini 1 j=1 mi! jal m! 22 121022! Am I Zaj elearly, (Nisa Nm 2N =) - Mreltinomial (nj tis Pa, ai Tohere j=1()m. Σλή jal Pm) 9

mean beet sinee it is given that, therefore, P[N-nis. m n! In - – – Nm = 02m Ni=n intensity di = 1 *i=1c0) m (-) m! (2 and in-Zing) silna! ,n=o, 1,2,- then, since (No we assume on. Rim = nm 9 i=1 m. O m 1=1 n1 è ? (b). Given No Poisson (a), ve P(N= n) = n! P[N-ni, Nm = m N=n Nm) are random me. n- teeple, Without loss of generality Ni Poisson ( 11) &i=1 (1) m. P(N=R, m) 2 Ri, d = iri P(N=n) at Ini I P(Ni = ni ni) im P(N=R) ! .. n! n! -- Am! rohich is same as previous The unconditional dist" of (Nis - m) is same as the conditional dist" of (N .nm) given ZiNi = because the disth remains invariant render the condition of Ni's. (9)". a .1m