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Suppose th at X1,...,X, is a random sample from the G eomet ric distribution with parameter 0 € (0, 1 ), Geo(0), with probability mass function Px (x; 0) = 0(1 – 0)*-1, x € {1,2, ...}.

Suppose th at X1,...,X, is a random sample from the G eomet ric distribution with parameter 0 € (0, 1 ), Geo(0), with probability mass function Px (x; 0) = 0(1 – 0)*-1, x € {1,2, ...}.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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2. Suppose that X1,..X, is a random sample from the Geomet ric distribution with parameter 0 € (0,1), Geo(0), with probability mass function Px(x; 0) = 0(1 – 0)*- xe {1,2...}. Recall that E[X;] = 1/0. (a) Suppose Y has a Negative Binomial distribution with paramet ers n e N and PE (0, 1), NB(n, p), with probability mass function Py p°( .... n - Show that the probability generating fun ction of Y is zp Gy(z) 1- (1 – p)z, (b) Show using generating fun ctions or otherwise that , X; ~ NB(n, 0). (c) Show that the Geometric distribution is an expon en tial family and det ermin e the can on ical statistic bas ed on the above random sample of size n.
Transcribed Image Text:2. Suppose that X1,..X, is a random sample from the Geomet ric distribution with parameter 0 € (0,1), Geo(0), with probability mass function Px(x; 0) = 0(1 – 0)*- xe {1,2...}. Recall that E[X;] = 1/0. (a) Suppose Y has a Negative Binomial distribution with paramet ers n e N and PE (0, 1), NB(n, p), with probability mass function Py p°( .... n - Show that the probability generating fun ction of Y is zp Gy(z) 1- (1 – p)z, (b) Show using generating fun ctions or otherwise that , X; ~ NB(n, 0). (c) Show that the Geometric distribution is an expon en tial family and det ermin e the can on ical statistic bas ed on the above random sample of size n.
(d) Show that X,1x,=3 is an unbiased estimat or of (1 – 0)², wh ere for event A, 14 is the random variable defined as [1 if w e A 0 otherwis e. 14(w) (e) For each t > n + 2, and x € {1, ..., t -n - 1}, show that P (X,1x,=3 = X n-3 = (E) i=1 n-1 Hint: notice that for each x e {1, 2,...}, {X,1x,=3 = x} = {X1 = x, X, = 3}. (f) Hen ce or otherwise find the unique min imum variance un bias ed estimat or of (1 – 0)?. Hint: you may use without proof tーn-1 t - x - 4 (t - n)(t – n- 1) (t - 1)(t – 2) n - 3
Transcribed Image Text:(d) Show that X,1x,=3 is an unbiased estimat or of (1 – 0)², wh ere for event A, 14 is the random variable defined as [1 if w e A 0 otherwis e. 14(w) (e) For each t > n + 2, and x € {1, ..., t -n - 1}, show that P (X,1x,=3 = X n-3 = (E) i=1 n-1 Hint: notice that for each x e {1, 2,...}, {X,1x,=3 = x} = {X1 = x, X, = 3}. (f) Hen ce or otherwise find the unique min imum variance un bias ed estimat or of (1 – 0)?. Hint: you may use without proof tーn-1 t - x - 4 (t - n)(t – n- 1) (t - 1)(t – 2) n - 3
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