ST202 Probability Distribution Theory and
Inference Week 3: Exercises
1. Let Y = (Y1;:::;Yn) be a random sample from a N( ; 2) population.
(a) Show that the joint density of the sample Y is
fY (y1;:::;yn) = (2 2) n=2 exp
Pn i=1(Yi ) 2 2 2
(b) Show that the above is equal to
fY (y1;:::;yn) = (2 2) n=2 exp
n( y ) 2 + (n 1)S2 2 2
2. Find a su cient statistic for the case of a random sample of size n from a
Bernoulli(p) distribution, using (i) the de nition of a su cient statistic and (ii)
the factorization theorem.
3. Find a su cient statistic for the case of a random sample of size n from
(a) Binomial with parameter p, 0 , and
fY (yj ) = 0 for y .
4. Let Y1;Y2;:::;Yn be a random sample from a Gamma( ; ) distribution with
pdf
fYi(yi) = y
1
i
( )
e yi; ; > 0; yi > 0
Find the method of moments estimators of and