Let (X1, X2, . .., Xn) denote a random sample from the density function '(0

Let (Xl, . Xn) denote a random sample from the density function f (TIO) where 0 G (0, 1) is an unknown parameter. Denote Nj to be the number ofobservations among the sample elements which belong to the interval , , = 1, 2andN3 = nN1 N2. For confidentiality, only summaries of the sample are released (31 to two analysts, such that Analyst A knows only (NI , N2 , N3) and Analyst B knows only (NI , N2-FN3). QI. Show that (NI, N2, N3) follows a multinomial distribution with probabilities I 9, O) and 02, respectively (S marks). Note that a random variable (X, Y, Z) follows a multinomial distribution with size n and proba- bilities p, q, andr 1 p q if X -4- Y + Z = n and their joint p.m.f. is a!b!c! for all non-negative integers a, b, c such that a -F b + c = n