14.381: Statistics
Problem Set 1
1. Let X and Y be random variables with finite variances. Show that
min E (Y − g(X))2 = E (Y − E (Y | X))2 ,
g(·)
where g(·) ranges over all functions.
2. Let X have Poisson distribution, with parameter θ
X ∼ Poisson(θ) ⇐⇒ P {X = j} =
e−θ θj
j!
j = 0, 1, ...
Let Y ∼ Poisson(λ) be independent on X.
(a) Show that X + Y ∼ Poisson(λ + θ).
(b) Show that X|X + Y is binomial with success probability
θ
.
θ+λ
Note: Variable ξ has binomial distribution with success probability p and
parameter n if
P {ξ = j} =
n!
pj (1 − p)n−j ;
j!(n − j)!
j = 0, 1, . . . , n
3. Show that if a sequence of random variables ξi converges in distribution to a
p
constant c, then ξi → c.
4. Let {Xi } be independent Bernoulli (p). Then EXi = p, V ar(Xi ) = p(1 − p).
P
Let Yn = n1 ni=1 Xi .
(a) Show that
√
n(Yn − p) ⇒ N (0, p(1 − p)).
6
(b) Show that for p =
1
2
the estimated variance Yn (1 − Yn ) has the following
limit behavior
√
n(Yn (1 − Yn ) − p(1 − p)) ⇒ N (0, (1 − 2p)2 p(1 − p)).
1
(c) Show that for p =
1
2
·
¸
1
1
n Yn (1 − Yn ) −
⇒ − χ21
4
4
χ21 is a chi-square distribution with 1 degree of freedom.
P
ξ1 , . . . , ξp be i.i.d. N (0, 1), then χ2p = pi=1 ξi2 .
Note:
Let
Curious fact: Note that Yn (1 − Yn ) ≤ 14 , that is, we always underestimate
the variance for p = 21 .
(d) Prove that if (i)
√
n
σ
(ξn − µ) ⇒ N (0, 1) (ii) g is twice continuously differ-
entiable: g 0 (µ) = 0, g 00 (µ) 6= 0, then
n(g(ξn ) − g(µ)) ⇒ σ 2
g 00 (µ) 2
χ1 .
2
Note. You may assume that g has more derivatives, if it simplifies your life.
5. Let X1 , X2 , ..., Xn ∼ i.i.d. N (µ, σ 2 ). Let us define
n
n
1X
1 X
2
X=
X i , sX
=
(Xi − X)2 ;
n i=1
n − 1 i=1
and
k
Yj =
(a) Show that
k
Xj − µ
1X
1 X
;Y k =
Yi ; s2k =
(Yi − Y k )2 .
σ
k i=1
k − 1 i=1
(n−1)s2X
σ2
= (n − 1)s2n .
(b) Check that
(k − 1)s2k = (k − 2)s2k−1 +
¢2
k−1¡
Yk − Y k−1 .
k
2
(c) Prove that if (k − 2)s2k−1 ∼ χk−2
then (k − 1)s2k ∼ χ2k−1 .
(d) Check that s22 ∼ χ12 .
(e) Conclude that
n−1 2
s
σ2 X
∼ χ2n−1 .
Hint: You can use the fact proved on the lecture that random variables s2k and
Y k are independent.
2
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14.381 Statistical Method in Economics
Fall 2013
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