Question

2. (Long-tailed and short-tailed random variable) [10]

A long-tailed RV X is a random variable satisfying ∀t > 0,

lim s→∞ Pr{X > s + t|X > s} = 1.

A short-tailed RV X is a random variable satisfying ∀t > 0,

lim s→∞ Pr{X > s + t|X > s} = 0.

From Problem 1, we see that an exponential RV X ∼ Exp(λ) is neither long-tailed nor shorttailed because

Pr{X > s + t|X > s} = e−λt, ∀s,t > 0.

a) Consider a Cauchy random variable X whose PDF is given by
fX(x) =
c /(1 + x2)
, x ∈R.
for some constant c.

• Find the constant c. [2]

• Find the distribution function of X. [2]

• Show that a Cauchy random variable is long-tailed. [2]

b) Show that a standard Gaussian RV is short-tailed. [4]

The complementary CDF is also called the (right) tail probability.

Note that Pr{X > s + t|X > s} = Pr{X > s + t}/Pr{X > s}.

For a long-tailed/short-tailed RV, its complementary CDF goes to zero at relative slow/fast speed.

Answer 1