Homework problems for MATH 5075/6820
Deadline 29 January
Problem 1 Let X and Y be independent and identically distributed exponential random variables
with EX = EY = 1. Compute the distribution function of Z = X + Y .
Solution. By the convolution formula,
h(t) =
∫ ∞
−∞
f(t− u)f(u)du =
∫ ∞
0
f(t− u)e−udu =
∫ t
0
e−(t−u)e−udu = te−t
if t > 0 and the density is 0 if t ≥ 0. The exponential density is
f(t) = e−tI{t ≥ 0}.
Problem 2∗ Let X, Y and Z be independent and identically distributed exponential random
variables with EX = EY = EZ = 1. Compute the distribution function of V = ZX+Y .
Solution. The density of Z is exponential
f(t) = e−tI{t ≥ 0}
and the density of X + Y is Gamma(1)
g(t) = te−tI{t ≥ 0}.
Now x > 0
P {V ≤ x} =
∫ ∞
0
P
{
Z
t
≤ x
}
g(t)dt =
∫ ∞
0
P{Z ≤ tx}te−tdt =
∫ ∞
0
(1− e−tx)te−tdt
and ∫ ∞
0
(1− e−tx)te−tdt =
∫ ∞
0
te−tdt−
∫ ∞
0
te−t(x+1)dt∫ ∞
0
te−tdt = 1 (the integral of a density is 1)
Using the density of a Gamma(1, 1/(x+ 1)) and the notation λ = 1/(1 + x)∫ ∞
0
te−t(x+1)dt =
1
1 + x
∫ ∞
0
t
λ
e−t/λdt =
1
1 + x
The derivative is the density
fV (x) =
1
(1 + x)2
I{x > 0}.
Problem 3 Let X1, X2, . . . , Xn be independent and identically distributed random variables with
distribution function
F (x) =
0, x < 0
1− 1
1 + x
, x ≥ 0.
Let Xn,n = max1≤i≤nXi and Yn = Xn,n/n. Show that Yn converges in distribution and determine
the limit.
Solution.
P {Yn ≤ x} = P
{
max
1≤i≤n
Xi ≤ xn
}
= Fn(xn)
2
and for x > 0
Fn(xn) =
(
1− 1
1 + nx
)n
=
(
nx
1 + nx
)n
=
(
1 + nx
nx
)−n
=
(
1 +
1/x
n
)−n
=
[(
1 +
1/x
n
)n]−1
→ e−1/x
The limit distribution is 0 if x < 0. We used the result(
1 +
y
n
)n
→ ey for all y
Problem 4∗ Let X1, X2, . . . , Xn be independent and identically distributed random variables with
distribution function
F (x) =
0, x < 0
1− 1
1 + x
, x ≥ 0.
Let Fn(t) be the distribution function of Yn = max1≤i≤nXi/n. Show that there is a distribution
function G and a constant C such that
sup
1≤x<∞
|Fn(x)−G(x)| ≤
C
n
,
where G denotes the limiting distribution function.
S
¯
olution. We use the distribution of Yn from Problem 3. We also use
| log(1 + h)− h| ≤ h2/4, if |h| ≤ h0
and ∣∣∣e−x − e−x+h∣∣∣ ≤ |h|
for all x > 0. Now (
1 +
1/x
n
)−n
= exp
(
−n log(1 + 1/x
n
)
)
and since we can assume 1/(nx) ≤ 1/n ≤ h0
exp (−1/x− 1/(4n)) ≤ exp
(
−n log
(
1 +
1/x
n
))
≤ exp (−1/x+ 1/(4n))
and ∣∣∣exp (−1/x− 1/(4n))− e−1/x∣∣∣ ≤ 1
4n
,∣∣∣exp (−1/x+ 1/(4n))− e−1/x∣∣∣ ≤ 1
4n
.
Deadline 5 Fe
uary
Problem 5 Let X1, X2, . . . , XN be independent and identically distributed random vectors with
EXi = 0 and EX
2
i = 1. Compute
E
(
n∑
i=1
1
i
Xi
)2
and show ∣∣∣∣∣
n∑
i=1
1
i
Xi
∣∣∣∣∣ = OP (1).
Problem 6∗. Let (Xi, Yi), 1 ≤ i ≤ n be independent and identically distributed random variables
3
with EXi = EYi = 0, EX
2
i = σ
2
1, EY
2
i = σ
2
2 and co
(Xi, Yi) = ρ. Compute
E
(
n∑
i=1
Zi
)2
,
where
Zi = Xi +
1
i
Yi, 1 ≤ i ≤ n.
Problem 7. Let X1, X2, . . . , Xn be a sequence of random variables with EXi = 0,
EXiXj =
σ2, i = j,
ρ |i− j| = 1,
0, |i− j| > 1.
Compute
E
(
n∑
i=1
Xi
)2
and show that
1
n
n∑
i=1
Xi
P→ 0.
Problem 8∗. Let X1, X2, . . . , Xn be independent and identically distributed exponential random
variables with EXi = 1. Compute
E
(
1
X1 +X XXXXXXXXXXXn
)
.
Deadline 12 Fe
uary
Problem 9 Let �i be a sequence of random variables with E�i = 0, E�
2
i = σ
2 and E�i�j = 0 if
i 6= j. Let x0 = 0 and |ρ| > 1. The sequence xk defined by
xi = ρxi−1 + �i, i = 1, 2, . . . .
Compute Exk and Exkxk+2.
Problem 10 Let �i,−∞ < i < ∞, be independent and identically distributed random variables
with �i = 0 and �
2
i = σ
2. Let xi be the stationary solution of
xi =
1
3
xi−1 + �i, −∞ < i <∞.
If var(x0) = 100, determine σ
2.
Problem 11* Let �i,−∞ < i < ∞, be independent and identically distributed normal random
variables with �i = 0 and �
2
i = σ
2. Let xi be the stationary solution of
xi = ρxi−1 + �i, −∞ < i <∞,
|ρ| < 1. Compute the joint distribution of xk and x`.
4
Problem 12* Let �i,−∞ < i < ∞, be independent and identically distributed random variables
with �i = 0 and �
2
i = σ
2 and define
x0 =
∞∑
`=0
c`�−`.
Prove that x0 is well defined. i.e. the infinite sum defining x0 converges with probability 1 if
∞∑
`=0
c2` <∞.
Deadline 19 Fe
uary
Problem 13 Let �i,−∞ < i < ∞ be independent and identically distributed random variables
with E�i = 0 and E�
2
i = σ
2. Write the stationary solution of
xi = 0.7xi−1 − 0.10x2 + �i
in a causal for. i.e. an infinite sum of the {�j , j ≤ i}.
Problem 14 Let �i,−∞ < i < ∞ be independent and identically distributed random variables
with E�i = 0 and E�
2
i = σ
2. Write the stationary solution of
xi = −0.25x2 + �i
in a causal for. i.e. an infinite sum of the {�j , j ≤ i}.
Problem 15* Let �i,−∞ < i < ∞ be independent and identically distributed random variables.
Let xi be the stationary causal solution of
xi = φ1xi−1 + φ2xi− XXXXXXXXXXφpxi−p + �i.
Prove that E|xi|ν <∞ if and only if E|�0|ν <∞, where ν > 0.
Problem 16* Let �i,−∞ < i < ∞ be independent and identically distributed random variables,
E�i = 0 and E�
2
i = σ
2. Show that, if |φ1|+ |φ2| < 1, then
xi = φ1xi−1 + φ2xi−2 + �i
has a stationary causal solution.