Page 85 - Probability and Statistical Inference

62 1. Notions of Probability

θ and β

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where θ ∈ ℜ and β ∈ℜ . Here, θ and β are referred to as parameters. In the
statistical literature, this is known as the Laplace or double exponential distri-
bution. The Figure 1.8.2 gives a plot of this pdf when θ = 0 and β = 1, 2. It is
implicit that x, f(x) are respectively plotted on the horizontal and vertical axes.
(i) Show that the pdf f(x) is symmetric about x = θ
(ii) Derive the expression of the df explicitly;
(iii) Show that P{| X – θ| > a + b | |X – θ| > a} does not depend on
a, for any two positive numbers a and b;
(iv) Let θ = 0 and β = 1. Obtain an expression for the right hand tail
area probability, P(X > a), a > 0. Then compare tail area prob-
ability with that of the Cauchy distribution from (1.7.31). Which
distribution has “heavier” tails? Would this answer change if θ =
0 and β = 2 instead? {Hint: Recall the types of discussions we
had around the Table 1.7.1.}
1.7.14 (Exercise 1.6.7 Continued) Consider a random variable X having the
pdf f(x) = c(x – 2x + x )I(0 < x < 1) where c is a positive number and I(.) is the
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indicator function. Find the value of c. {Hint: In order to find c, should you
match this f(x) up against the Beta density?}
1.7.15 Show that
0) given by (1.7.27) is a bona-fide pdf with µ ∈ ℜ, σ ∈ ℜ . Also verify that
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(i) P(X ≤ x) = Φ({log(x) – µ}/σ), for all x ∈ ℜ ;
(ii) P{log(X) ≤ x} = Φ({x – µ}/σ), for all x ∈ ℜ.
{Hint: Show that (a) f(x) is always positive and (b) evaluate by
making the substitution u = log(x).}
1.7.16 The neighborhood fishermen know that the change of depth (X,
measured in feet) at a popular location in a lake from one day to the next is a
random variable with the pdf given below:

where c is a positive number. Then,
(i) find the value of the constant c;
(ii) determine the expression of the df;
(iii) find the median of this distribution.
1.7.17 The neighborhood fishermen know that the change of depth (X,
measured in feet) at a popular location in a lake from one day to the next