Page 253 - Probability and Statistical Inference

230 4. Functions of Random Variables and Sampling Distribution

U will be zero when u ≤ 0 or u ≥ 2, but it will be positive when 0 < u < 2.
Thus, using the convolution result, for 0 < u < 1, we can write the df of U as
. Hence, for 0 < u
< 1 we have . On the other hand, for 1 ≤ u < 2, we can

write the df of U as
so that
4.2.10 Suppose that X and X are independent random variables, distrib-
2
1
uted uniformly on the intervals (0, 1) and (0, 2) with the respective pdf’s
f (x ) = I(0 < x < 1) and f (x ) = 1/2I(0 < x < 2). Use the convolution result
2
2
1
2
1
1
to derive the form of the pdf g(u) of U = X + 1/2X with 0 < u < 2.
2
1
4.2.11 (Example 4.2.10 Continued) Let U, V be iid N(µ, σ ). Find the pdf
2
of W = U + V by the convolution approach. Repeat the exercise for the
random variable T = U – V.
4.2.12 (Example 4.2.11 Continued) Provide all the intermediate steps in the
Example 4.2.11.
4.2.13 Prove a version of the Theorem 4.2.1 for the product, V = X X , of
1 2
two independent random variables X and X with their respective pdf’s given
2
1
by f (x ) and f (x ). Show that the pdf of V is given by h(v) =
1
1
2
2
for v ∈ ℜ. Show that h(v) can be equivalently
written as . {Caution: Assume that V can not take
the value zero.}
4.2.14 Suppose that X , X , X are iid uniform random variables on the
1
3
2
interval (0, 1).
(i) Find the joint pdf of X , X , X ;
3:1 3:2 3:3
(ii) Derive the pdf of the median, X ;
3:2
(iii) Derive the pdf of the range, X – X .
3:1
3:3
{Hint: Use the Exercise 4.2.7}
4.2.15 Suppose that X , ..., X are iid N(0, 1). Let us denote Y = 4Φ(X )
i
n
i
1
where .
(i) Find the pdf of the random variable
(ii) Evaluate E[U] and V[U];
(iii) Find the marginal pdf’s of Y , Y and Y – Y .
n:1 n:n n:n n:1
4.2.16 Suppose that X is uniform on the interval (0, 2), X has its pdf 1/
1
2
2
2xI(0 < x < 2), and X has its pdf 3/8x I(0 < x < 2). Suppose also that X , X ,
3
2
1
X are independent. Derive the marginal pdf’s of X and X . {Hint: Follow
3:1
3:3
3
along the Example 4.2.8.}