Page 50 - Elements of Distribution Theory

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36 Properties of Probability Distributions

1.14 Let F 1 and F 2 denote distribution functions on R. Which of the following functions is a distri-
bution function?
(a) F(x) = αF 1 (x) + (1 − α)F 2 (x), x ∈ R, where α is a given constant, 0 ≤ α ≤ 1
(b) F(x) = F 1 (x)F 2 (x), x ∈ R
(c) F(x) = 1 − F 1 (−x), x ∈ R.
1.15 Let X denote a real-valued random variable with an absolutely continuous distribution with
density function
2x
p(x) = , 0 < x < ∞.
(1 + x )
2 2
Find the distribution function of X.
1.16 Let X denote a real-valued random variable with a discrete distribution with frequency function
1
p(x) = , x = 1, 2,....
2 x
Find the distribution function of X.
1.17 Let X 1 , X 2 ,..., X n denote independent, identically distributed random variables and let F
denote the distribution function of X 1 . Define
1 n
ˆ F(t) = I {X j ≤t} , −∞ < t < ∞.
n
j=1
Hence, this is a random function on R.For example, if denotes the sample space of the
underlying experiment, then, for each t ∈ R,
1 n
ˆ F(t)(ω) = I {X j (ω)≤t} ,ω ∈ .
n
j=1
Show that ˆ F(·)isa genuine distribution function. That is, for each ω ∈ , show that ˆ F(·)(ω)
satisfies (DF1)–(DF3).
1.18 Let X denote a nonnegative, real-valued random variable with an absolutely continuous distri-
bution. Let F denote the distribution function of X and let p denote the corresponding density
function. The hazard function of the distribution is given by
p(x)
H(x) = , x > 0.
1 − F(x)

(a) Give an expression for F in terms of H.
(b) Find the distribution function corresponding to H(x) = λ 0 and H(x) = λ 0 + λ 1 x, where
λ 0 and λ 1 are constants.
1.19 Let X denote a real-valued random variable with a Pareto distribution, as described in Exam-
ple 1.28. Find the quantile function of X.
1.20 Let X denote a real-valued random variable and let Q denote the quantile function of X. Show
that Q(.5) is a median of X.
1.21 Let X 1 and X 2 denote real-valued random variables such that, for j = 1, 2, the distribution of
X j has a unique median m j . Suppose that Pr(X 1 > X 2 ) > 1/2. Does it follow that m 1 ≥ m 2 ?
1.22 Let F 1 and F 2 denote distribution functions for absolutely continuous distributions on the real
line and let p 1 and p 2 denote the corresponding density functions. Which of the following
functions is a density function?
(a) αp 1 (αx) where α> 0
α
(b) p p 1−α where 0 ≤ α ≤ 1
1 2
(c) αp 1 + (1 − α)p 2 where 0 ≤ α ≤ 1