6More distributions
IA Probability
6.1 Cauchy distribution
Definition (Cauchy distribution). The Cauchy distribution has pdf
f(x) =
1
π(1 + x
2
)
for −∞ < x < ∞.
We check that this is a genuine distribution:
Z
∞
−∞
1
π(1 + x
2
)
dx =
Z
π/2
−π/2
1
π
dθ = 1
with the substitution x = tan θ. The distribution is a bell-shaped curve.
Proposition. The mean of the Cauchy distribution is undefined, while
E
[
X
2
] =
∞.
Proof.
E[X] =
Z
∞
0
x
π(1 + x
2
)
dx +
Z
0
−∞
x
π(1 + x
2
)
dx = ∞ − ∞
which is undefined, but E[X
2
] = ∞ + ∞ = ∞.
Suppose
X, Y
are independent Cauchy distributions. Let
Z
=
X
+
Y
. Then
f(z) =
Z
∞
−∞
f
X
(x)f
Y
(z − x) dx
=
Z
∞
−∞
1
π
2
1
(1 + x
2
)(1 + (z −x)
2
)
dx
=
1/2
π(1 + (z/2)
2
)
for all
−∞ < z < ∞
(the integral can be evaluated using a tedious partial
fraction expansion).
So
1
2
Z
has a Cauchy distribution. Alternatively the arithmetic mean of
Cauchy random variables is a Cauchy random variable.
By induction, we can show that
1
n
(
X
1
+
···
+
X
n
) follows Cauchy distribution.
This becomes a “counter-example” to things like the weak law of large numbers
and the central limit theorem. Of course, this is because those theorems require
the random variable to have a mean, which the Cauchy distribution lacks.
Example.
(i)
If Θ
∼ U
[
−
π
2
,
π
2
], then
X
=
tan θ
has a Cauchy distribution. For example,
if we fire a bullet at a wall 1 meter apart at a random random angle
θ
, the
vertical displacement follows a Cauchy distribution.
1
θ
X = tan θ
(ii) If X, Y ∼ N(0, 1), then X/Y has a Cauchy distribution.