Transcribed Text
Part A
1. Consider n rolls of a balanced die. Let X, be the outcome of the ith roll and let
8.=
(a) Find an asymptotic approximation (large n) to the distribution of X
(b) Find c, such that 8.
2. Let X14 < X24 < X34 < Xes be the order statistics of a random sample of size 4 from
the distribution having pdf f(x) zero elsewhere. Find P(3: < Xes).
3. Let Y denote the nth order statistic of a random sample of size n from a distribution
having pdf, f(x)  1/0, and zero elsewhere. Find the
limiting distribution of Xnn
4.
Let X,  Gamma(e, N  n), i = 2, n where @ does not depend on n. Find an
asymptotic approximation (large n) to the distribution of T.  Xa/n
5. Let of be the sample variance of n independent random variables, ench distributed as
02). Prove that & converges in probability to of
Part B
1. Let Y,  xi.) and
2120  22
Show that z,Azz Normal(0,1).
2. Let x  Beta(a,6), where a, ,6>0. Derive E(X").
3. Let X,  Normal(a, and define U  and W 
(a) Find a statistics that is a function of U and W that is unbiased for 02 + 12.
(b) Let c be a constant, and define Y,  1 if X, s c and zero otherwise. Find a
statistic that is a function of Y1.Y2. Y. and also unbiased for Fx(c)  (+)
B11 N (n.) p)
BIN(I,p)
pri  p)
np. np(1p)
(pe'+9)"
NB (r.p)
CAN
GEO(p)
prot
r/prain?
1/p.g/p²
(F)
in
18
HY P(n,M,N)
CON/C)
POI(A)
cape
enter1)
UNI F(a,b)
1/(ba)
N 02)

agb.
it
11,02
GAMMA (0,k)
EXP(#)
no
0,0°
(Ita)"
Ft
EX. P(0,7)
DE (0.v)
se
noto.or
ipe

WEI (0.8)
towa, exp  (x/0)
EV (0,n)
 
@r(1 + 3),ugly
e"T(1+01
CAU (0,1))
PAR (0,x)
**
x2(v)
t (t)
(I+E)
22
**
M/2
BETA (a, b)
l'(a+b)
Theorem (2.4.7 Chebychev]: If X is random variable with mean II and variance 02, then for
any * > 0, PIIX " > ka) s 1/N which can also be written as <ka)>11/k2. 
</ka)>
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