Download HOMEWORK I: PREREQUISITES FROM MATH 727 Question 1. Let

HOMEWORK I: PREREQUISITES FROM MATH 727
Question 1. Let X1 , X2 , . . . be independent exponential random variables with mean µ.
(a) Show that for n ∈ Z+ , we have EX1n = µn n!.
(b) Show that almost surely,
X1 + · · · + Xn
√
→∞
n
(c) Find the limit in distribution of
Yn =
X1 + · · · + Xn − nµ
√
n
Zn =
X1 + · · · + Xn − nµ
p
.
X14 + · · · + Xn4
and
Question 2. Let U1 , U2 , . . . , be independent random variables that are
uniformly distributed in (0, 1). Let
T := inf {n ≥ 2 : U1 + · · · + Un > 1} .
(a) Let a ∈ [0, 1]. Show that
an
.
n!
(b) Show that for any integer-valued random variable X ≥ 0, we have
P(U1 + · · · + Un ≤ a) =
EX =
∞
X
P(X ≥ n).
n=1
(c) Show that ET = e ≈ 2.71.
Question 3. Let X1 , . . . , Xn be i.i.d. Bernoulli random variables with
parameter p ∈ (0, 1). Let
n
1 X 2
Xi
T :=
n i=1
n
X 2
1
T :=
−
X
+
Xi ,
j
(n − 1)2
i=1
j
and
n
n−1X j
J := nT −
T .
n j=1
(a) Show that T converges to p2 in probability as n → ∞.
(b) Show that EJ = p2 .
Question 4. Let T = X1 + · · · + Xn , where the Xi are i.i.d. Poisson
random variables with mean µ.
(a) Show that T is also a Poisson
random variable.
(b) Compute E E(X1 X2 | T )
(c) Fix k ≥ 0 and let t ≥ 0. Compute φ(t) := E(1[X1 = k]| T = t).
(d) Show that limn→∞ φ(T ) = P(X1 = k) almost surely.
Question 5.
(a) Let X have pmf given by P(X = a) = 61 , P(X = b) = 26 and
P(X = c) = 36 . Suppose I roll a three sided dice with distribution
X ten times. What is the probability that I get two a’s, three b’s
and five c’s?
(b) Let U1 , . . . , Un be independent random variables that are uniformly
distributed in [0, 1]. Order the random variables so that
V1 < V2 < · · · < Vn ,
where
{V1 , . . . , Vn } = {U1 , . . . , Un } .
Recall that here V1 , . . . , Vn are called the order statistics for U1 , . . . , Un .
For h small and x ∈ [0, 1) show that
n!
P(x ≤ Vk ≤ x+h, Vk−1 ≤ x, Vk+1 ≥ x+h) =
xk−1 (1−x−h)n−k h.
(k − 1)!(n − k)!
(c) Show that the pdf for Vk is given by a Beta distribution. Hint you
may use the fact that
|P(x ≤ Vk ≤ x + h) − P(x ≤ Vk ≤ x + h, Vk−1 ≤ x, Vk+1 ≥ x + h)|/h → 0
as h → 0, since the difference corresponds to an event that there is
more than two points in the interval [x, x + h].
(d) Show that if F is the cdf of a random variable X, then F −1 (U1 )
has the same distribution as X. Here, and after, assume that F is
strictly increasing. Note that this result holds in general by defining
F −1 (y) := sup {x ∈ R : F (x) < y} .
(e) Let F be the cdf for a continuous random variable. Let Wi =
F −1 (Ui ). Let Z1 < Z2 < · · · < Zn be the order statistics for
W1 , . . . , Wn . Show that Zk = F −1 (Vk ).
(f ) Show that if X1 , . . . , Xn are independent continuous random variables with cdf F and pdf f and order statistics given by Y1 < Y2 <
· · · < Yn , then the pdf for Yk is given by
n!
gk (x) =
F (x)k−1 (1 − F (x))n−k f (x).
(k − 1)!(n − k)!
Question 6. Let (X1 , . . . , Xn ) be independent standard normal random
variables and let
n
1X
X̄ =
Xi
n i=1
denote the usual sample average.
(a) Compute the distribution of X̄ by using the change of variables:
y1 = x̄, y2 = x2 − x̄, . . . , xn − x̄;
that is find the distribution of
Y = (Y1 , . . . , Yn ),
where Y1 = X̄ and Yi = Xi − X̄ for i ∈ [2, n] to obtain the distribution of X̄. Of course you already know what the answer should
be, but using this method, as a bonus, you will find that X̄ is independent of (Y2 , . . . , Yn ).
(b) Is the previous part regarding the independence true if Xi are not
normal random variables?
Question 7. Suppose X ∼ N (µ, V ), where µ ∈ Rn and V ∈ Rn×n is
a positive definite symmetric covariance matrix, so that X has a nondegenerate multivariate normal distribution. Let A ∈ Rn×n be a matrix
with det(A) 6= 0. Think of X ∈ Rn×1 as a column vector. What is the
distribution of Y = AX?
Note that: The pdf for X is given by
1
f (x) = p
exp[− 12 (x − µ)t V −1 (x − µ)],
n
(2π) | det(V )|
where x ∈ Rn×1 , and t denotes matrix transposition.