Question

Suppose that N ~ Poisson(1), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M1: X|N = n ~ N(2* n,o2 = 1) • Model M2: X|N = n ~ N(2*n+n2,02 = 1) The only difference between My and M2 is the conditional mean function - the mean function is linear in M1 and quadratic in M2. a If M1 is true: Find E1[X] and V1[X] as a function of b If M2 is true: Find E2[X] and V2[X] as a function of a c Assume that 1 = 2 for the following: (a) If M1 is true: Find upper bound by such that P(X > 20) < bu (b) If M2 is true: Find upper bound b2 such that P(X > 20) < b2 (c) We observe X = 25. If we thought that My and M2 were equally likely before seeing any data, which model now seems more plausible?

Answer 1

Solection: Given that No Poisson (A) then P(N=R) - et , n =0,1,2,. n! = 40+ 1 Again F(x) = art (N) - A. • Model Ms: X/N=n' (21,6 ² = 1) • Model M₂: X/N=1 a N(zn+n", 6-1) ca) sf Mi is true, E(X) = ENTE,XIN (*IN-->) E:"(zw) (**»* N(2001)) - 2 EN(N) = 2a (Ans.) M(x) = vi [ECXIN)] + F (Voor (XIN)) V (2N) + EN (1) - 4 Var! (N) + 1 (Ans.) (6) If M₂ is true; E₂(x). Ez (EXIN (XIN=n)) EM (2N+N?) 25,"(0) + *(03) +(Vech (as) + CE * CNC))) 20t a+ x² = 2² +37 -(Ans) V=(x) = V[(X/N)] + Ez((XID)) va (21+N3) + E, NO (1) E"{(zn+w)"}-{E"(20+N2)3 + 1 Var (T) = E(12) - F?CT) F[9w2+w++ 4N3] - (4+39) + 1 - 4 E(N2) + E (N-1) + 4 E(N) - 14-992-64° +1 4(1+22) + (44+62+72+ 2) +4(a + 3x + 1) - 11-942-613+1 E(N²) = x²+37 E(N) = a +372 + (x1 - x^) + (6x² + 4x² - 6x²) E(N) = 44+673 +71?+^ +(49°++*% + 127-993) + ( 42 +2 +42 ) + 1 = 424 1422 +9 % +1 (Ans.)

(e) Assume 2 = 2. 21 20 1 = bi (2=2) (E(X)=2) (a) when me is true, P(x >20) - ) ( Markov Tnequality says. P(x>a) < ECO).) zepper bound, bi-) (b) Similarly, when M2 is true, P(x320) < (8) = 0.2. 2²432 -0.5 NA 20 be - wpper bound, b2 = 0.5 For My : P(x720) (e). We observe X = 25 > 20. Modette sagessene Now we compute P(x>20) with actual value a normal approximation TX-E(X) 20 - 2a > 42+1 12 P(Z > 16) $(5-33) FOR M₂: P(x720) = P(X-E(%) co (almost) PG > 20 - (0232) Var(x) 23+147°+91+1 2=2. Plata P(Z > 10F) = P(770-27) 3- (0.97) = 1-0·8314 - 0.4686 though both the probabilities are small; but on the basis of the values, Model M2 is plausibl more sible.

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