Question

Let X be a random variable which follows truncated binomial distribution with the following p.m.f. P(X=x) =((n|x)(p^x)(1−p)^(n−x))/(1−(1−p)^n), if x= 1,2,3,···,n.

•Find the moment generating function (m.g.f.) and the probability generating function(p.g.f.).

•From the m.g.f./p.g.f., and/ or otherwise, obtain the mean and variance. Show all the necessary steps for full credit.

Answer 1

Given that for a nandom variable x, P(x= x) n 3 2C-1, 2, Cx p² (1-p) 1- (-p)n ie X follows touncated Binomial districretion with truncation at o MGF of X can be obtained as رموده OC -1 Mx (t) - IE etx s ncx p² (1)n- etx (i-pon (-p) s ncx 1 - (-p)" 3=1 1 4= v=m) [MG (petaje [ " ;)*-- (using the (expansion of (lex) (-p) + pet 1-(1-P) 1- (1-pipe) - (-)" 1-(1-p) n ZEI Similarly, the PGF of x can be obtained as Px (s) Est E sxncx pl.-p)*-* 1-(-p)" (-p)" Ŝ 1-(1-6)* =1 (ps.) (-p)" 1-(1-6)”. ncx n 1 ² nCe (ps.) C 31-0 -1 1-P (-p)" +_ps 1 - (-p) (L-peps)" – (-p)" 1-(1-P)" ICS Scanned with CamScanner

n T = 1 77 Now, We want Æx and var x. so, consicha EX 2. Cx p? (1-p)"-x 1-(-p) (1-2) 3 x. » 1-(1-p)" 15 X! (on-2)! (1-0)” n(n-1)! L-11-p)" (x-1)! (0-2)! c EW - p 251 (h ne) C-p)” (6-1)! x-1 1-P (р x = (x4) (1-2)/(1-p n-1 min) (1-1" + P -P 1-(1-2) np. Scanned with (1-p)" fcs CamScanner H

In order to calculate variance, consider Ex(x-1). u E x(x-1) 2-) E 260-1) " (x b*(-p)»-x | -(i-p) x(x-1) n! (n-x)! x! X = (1-2) L-i-p) x=2 n 2-2 2 (1-2) 1-(-p) E n(n-1) (0-2)! (2-2)| (n-ad 22,64) L = 2 (1-5) n n(n-1)/6 1² (1-6)” 1-(i-p)22=2 QC-2)! (n-3 = n(n-1)/P /1 + P Delen 2 (7-2)! noires n-2 --16")* (=p)" d-1 1-(-p) n(n-1) p² 1-(-p) A But since JE XX1 JE X? - EX. => EX? - Ex(x-1) + Ex n(n-1) b? + ne L-(-p)". Thin. Var (x) : EXP-LIE x)__ nln-1) p’emp 1- (p) CS Scanned with CamScanner 2 (1 ne 1-6-b)

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