Question 1 (20 marks]
1. Let Y1 < Y2 < ···< Yu be the order statistics of 11 independently and identically distributed
continuous random variables X1 , X2 , ... , X11 with pdf f given by
Then find
fx(x)=}~, forO<x<3
lo, otherwise
= I.I. The pdf of the r th order statistics. Hint: fy/y)
(n-r)~~r-l)!
[Fx(y)y- 1 [1- Fx(y)]n-rfx(Y)
(3]
1.2. The pdfofthe minimum order statistics
13)
1.3. The pdf of the maximum order statistics
[3)
1.4.The pdf of the median
[3]
1.5.The joint pdf ofYv Y2 , ... , Yu
[31
1.6. Ifthe number ofrandom variables is reduced to 3, thus, X1 ,X 2 , X3 , then find the joint pdf of the
minimum and maximum order statistics.
[41
= Hint: fY1,y/y,, Yj) (i-l)!U-~~l)!(n-j)! [F(yJ]i-1 f (yi) [F(yj) - F(yi) f-i-1 f(Yj) [1 - F(yj)r-j
Question 2 (22 marks]
2.1. If the random variables Xi, ... , Xm are independent and if Xi has the x2 distribution with k
= degrees of freedom (i 1, ... , m), then show, using the moment generating function, that
the sum Y = X1 + . . . + Xm has the x2 distribution with km degrees of freedom. Hint:
k
use
Mx (t)
l
= (-1-21-t)
2.
[81
2.2. If z1, ... , z20 denote a random sample from a standard normal distribution, find the value of c
such that P(.Et1~zf $ c) = 0.25
[41
= = 2.3. Consider two independent samples of size n 1 8 and n 2 12 from two normal populations
with populations variances c,f= 3c,fT.hen, find the value b of such that Pe: ::;b)= 0.95.
(51
2.4. Let the random sample X1 ,X2 , ... , Xn ~N(µ, c, 2 ) where bothµ and c, 2 are unknown. Derive the
100(1 - a)% Cl forµ using the pivotal quantity method.
(51
Question 3 [25 marks]
3.1. Let X1 , X 2, •.. , Xn denote a random sample a Rayleigh distribution with parameter 0.
fx(xiJ0) = f20xie- 0xt2, for xi~ 0 and e > 0
lo,
otherwise
3.1.1. Find the maximum likelihood estimator of 0
(6)
3.1.2. Find the maximum likelihood estimator of g(0) = -le+ 2
(3)
3.1.3. Show that If=ixf is sufficient fore
(4)
3.2. Let 0 > 0 and let X 1 , ... , Xn be a random sample of size n from a distribution of pdf
f(x;
0)
=3
03
x
2
,
for
O< x < 0
Find the estimator of 0 using the method of moments.
(6)
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