Page 254 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 231

4.2.17 Suppose that Z is the standard normal random variable. Denote X =
max{|Z|, 1/|Z|}.
(i) Find the expression of P{X ≤ x} for x ≥ 1. Observe that P{X ≤
x} = 0 for x < 1;
(ii) Use part (i) to show that the pdf of X is given by f(x) = 2{φ(x)
–1
+ x φ(x )}I(x ≥ 1).
–2
–1
–1
{Hint: For x = 1, one can write P{X ≤ x} = P{x ≤ |Z|≤ x} = 2P{x ≤ Z =
–1
x} = 2{Φ(x) – Φ(x )}. This is part (i). Differentiating this expression with
respect to x will lead to part (ii).}
4.2.18 (Exercise 4.2.9 Continued) Suppose that X , X and X are inde-
1
3
2
pendent random variables, distributed uniformly on the interval (0, 1) with the
respective pdf’s f (x ) = I(0 < x < 1), f (x ) = I(0 < x < 1) and f (x ) = I(0
1
2
3
2
3
1
1
2
< x < 1). Use the convolution result to derive the form of the pdf h(v) of V =
3
X + X + X with 0 < v < 3. {Hint: Note that V = U + X where U = X + X 2
2
3
1
3
1
and its pdf g(u) was derived in the Exercise 4.2.9 for 0 < u < 2. Now, use the
convolution result again.}
4.2.19 (Example 4.2.9 Continued) Suppose that X , X and X are inde-
2
1
3
pendent random variables, distributed exponentially with the respective pdf’s
+
+
+
f (x ) = e I(x ∈ ℜ ), f (x ) = e I(x ∈ ℜ ) and f (x ) = e I(x ∈ ℜ ). Use
–x
–x
–x
2
2
3
3
3
2
3
2
1
1
1
1
the convolution result to show that the pdf h(v) of V = X + X + X with v ∈
3
1
2
+
ℜ matches with that of the Gamma(3, 1) distribution. {Hint: Note that V = U
+ X where U = X + X and its pdf g(u) was derived in the Example 4.2.9 for
3
1
2
u ∈ ℜ . Now, use the convolution result again.}
+
4.2.20 Suppose that X , ..., X are iid Uniform(0, 1) random variables
n
1
where n = 2m + 1 for some positive integer m. Consider the sample median,
that is U = X n:m+1 which is the order statistic in the middle.
m
m
(i) Show that the pdf of U is given by g(u) = cu (1 – u) × I(0 < u
2
< 1) where c = (2m + 1)!/(m!) . Is this one of the standard
distributions listed in the Section 1.7?
(ii) Show that E(U) = 1/2 and .
4.3.1 Suppose that X , ..., X are iid Poisson(λ). Use the mgf technique to
n
1
show that has the Poisson(nλ) distribution. {Hint: Follow along
the Examples 4.3.1-4.3.2.}
4.3.2 Suppose that X , ..., X are iid Geometric(p), 0 < p < 1. Find the
1 k
distribution of . The distribution of U is called Negative Binomial
with parameters (k, p). A different parameterization was given in (1.7.9).
4.3.3 Complete the arguments in the Example 4.3.5.