Page 251 - Probability and Statistical Inference

228 4. Functions of Random Variables and Sampling Distribution

4.2.5 In general, suppose that X , ..., X are iid continuous random variables with the
n
1
common pdf and df respectively given by f(x) and F(x). Derive the joint pdf of the i th
th
order statistic Y = X and the j order statistic Y = X , 1 ≤ i < j ≤ n by solving the
i
j
n:j
n:i
following parts in the order they are given. Define
u ), U = n – U – U . Now, show
that 2 3 1 2
(i) (U , U , U ) is distributed as multinomial with k = 3, p = F(u ),
1
1
1
3
2
p = F(u ) – F(u ), and p = 1 – p – p = 1 – F(u );
2
1
2
2
1
2
3
(ii) P{Y ≤ u ∩ Y ≤ u } = P{U ≥ i ∩ (U + U ) ≥ j};
i 1 j 2 1 1 2
(iii)
U = l} + P{U ≥ j};
2
(iv) the expression in part (iii) can be rewritten as
(v) the joint pdf of Y , Y , denoted by f(y , y ), can be directly
i j i j
obtained by evaluating first,
and then simplifying it.
4.2.6 Verify (4.2.7). {Hint: Use the Exercise 4.2.5.}
4.2.7 Suppose that X , ..., X are iid uniform random variables on the
1
n
interval (0, θ), θ ∈ ℜ. Denote the i order statistic Y = X where X = X n:2
th
i
n:i
n:1
≤ ... ≤ X . Define U = Y – Y , V = 1/2(Y + Y ) where U and V are
n
1
1
n:n
n
respectively referred to as the range and midrange for the data X , ..., X .
n
1
This exercise shows how to derive the pdf of the range, U.
(i) Show that the joint pdf of Y and Y is given by h(y , y ) = n(n –
n
1
1
n
n–2
1)θ (y – y ) I(0 < y < y < θ). {Hint: Refer to the equation
–n
1
n
n
1
(4.2.7).};
(ii) Transform (Y , Y ) to (U, V) where U = Y – Y , V = 1/2(Y +
n
1
n
n
1
Y ). Show that the joint pdf of (U, V) is given by g(u, v) = n(n – 1)
1
θ u when 0 < u < θ, 1/2u < v < θ – 1/2u, and zero otherwise;
n–2
–n
(iii) Show that the pdf of U is given by g (u) = n(n – 1)θ × (θ – u)
–n
1
n–2
u I(0 < u < θ). {Hint: g (u) = ,
1
for 0 < u < θ.};
(iv) When θ = 1, does the pdf of the range, U, derived in part (iii)
correspond to that of one of the standard random variables from
the Section 1.7?
4.2.8 Suppose that X , ..., X are iid uniform random variables on the
1
n
interval (θ, θ + 1), θ ∈ ℜ. Denote the i order statistic Y = X where
th
i n:i