1. Let X1, X2, . . .
iid⇠ Geometric(p). Let N ⇠ Poisson(), with N independent of all of the X 0is. Let
SN = X1 + . . .+XN .
(a) [2] Find P (SN = 0).
(b) [4] Find Cov(SN , N).
2. Let ⇢ 2 (1, 1), > 0, and ⌧ > 0. Suppose that Z ⇠ N(0, ⌧2), and X1 is a N(0,2) random variable
that is independent of Z. Let X2 = ⇢X1 + Z.
(a) [2] Find the joint distribution of X1 and X2.
(b) [1] State a condition so that X1 and X2 have the same marginal distributions if and only if
that condition is satisfied.
(c) [1] State a condition so that X1 and X2 are independent if and only if those conditions are
satisfied.
(d) [2] Show that if X1 and X2 are i.i.d., then P (X1 < X2) = 0.5.
3. Let U ⇠ Uniform(0, 1).
(a) [2] Let Xn be a discrete random variable with P (Xn =
i
n) =
1
n for i = 1, 2, . . . , n. Show that
Xn
d! U .
(b) [4] Find the asymptotic distribution of 2min(Xn, 1Xn).
4. Suppose that X1, X2, . . . , are iid with E[Xi] = µ and V ar[Xi] = 2 <1. Consider the sequence of
random variables Y1, Y2, . . . , where
Yn =
8><>:
14, if n = 1, 2, . . . , 1010,
1
n
nP
j=1
Xj , if n = 1010 + 1, 1010 + 2, . . .
.
(a) [2] What does Yn converge in probability to?
(b) [2] Find the asymptotic distribution of
p
n(Yn µ).
5. Let ✓ > 0 be a constant. Let X1, X2, . . . be i.i.d. with pdf f(x) =
(
✓x✓1, 0 < x < 1,
0, otherwise.
(a) [2] Find the distribution of log(X1).
(b) [2] Find the asymptotic distribution of
p
n
✓
1
n
nP
i=1
log(Xi) +
1
✓
◆
.
(c) [2] Find the asymptotic distribution of
p
n
0@1 + n
✓
nP
i=1
log(Yi)
1A.
(d) [2] Find the asymptotic distribution of
p
n
✓
nQ
i=1
Xi
◆ 1
n e 1✓
!
.
6. Let X1, . . . , Xn be i.i.d. Exponential(1/✓), and let Yi = 1/Xi for i = 1, 2, . . . , n.
(a) [2] Find the pdf of Yi for i = 1, 2, . . . , n.
(b) [2] Find the MLE of ✓.
(c) [1] Find the MLE of 1/✓.
7. Let W1, . . . ,Wn be i.i.d. positive r.v. where E[W
1
i ] exists and V ar[W
1
i ] <1.
(a) [2] What does
0@ nPi=1W1i
n
1A1 converge in probability to?
(b) [2] Find appropriate choices of an, b, and c such that an
0B@
0@ nPi=1W1i
n
1A1 b
1CA d! N(0, c).
8. Let t1, . . . , tn be known constants. Suppose thatX1, . . . , Xn are independent andXi ⇠ Exponential(1/✓i)
with log(✓i) = ti for all i = 1, 2, . . . , n.
(a) [2] Write down the equation that you would need to solve to find the maximum likelihood
estimator of .
(b) [2] Find the score function and the expected information function.
(c) [1] Let ˆ be the MLE of . Find appropriate choices of an, b, and c such that an(ˆb) d! N(0, c).
(d) [1] Find the MLEs of ✓ˆi for all i = 1, 2, . . . , n.
Summary of Discrete Distributions
Notation and
Parameters
Probability
Function
fŸx 
Mean
EŸX 
Variance
VarŸX 
Moment
Generating
Function
MŸt 
Discrete UniformŸa,b 
b ≥ a
a,b integers
1
b−a1
x a,a 1,… ,b
ab
2
Ÿb−a1 2−1
12
1
b−a1 ∑
xa
b
etx
t ∈ g
HypergeometricŸN, r,n 
N 1,2,…
n 0,1,… ,N
r 0,1,… ,N
r
xŸ   N−rn−xŸ  
N
nŸ  
x max Ÿ0,n − N r ,
… ,minŸr,n 
nr
N
nr
N Ÿ1 −
r
N  
N−n
N−1 Not tractable
BinomialŸn,p 
0 ≤ p ≤ 1, q 1 − p
n 1,2,…
n
xŸ  pxqn−x
x 0,1,… ,n
np npq Ÿpe
t q n
t ∈ g
BernoulliŸp 
0 ≤ p ≤ 1, q 1 − p
pxq1−x
x 0,1
p pq pe
t q
t ∈ g
Negative BinomialŸk,p 
0 p ≤ 1, q 1 − p
k 1,2,…
xk−1
xŸ  pkqx
−kxŸ  pkŸ−q x
x 0,1,…
kq
p
kq
p2
p
1−qet
k
t − lnq
GeometricŸp 
0 p ≤ 1, q 1 − p
pqx
x 0,1,…
q
p
q
p2
p
1−qet
t − lnq
PoissonŸ6 
6 ≥ 0
6xe−6
x!
x 0,1,…
6 6 e
6Ÿet−1 
t ∈ g
MultinomialŸn;p1,p2,… ,pk 
0 ≤ pi ≤ 1
i 1,2,… ,k
and∑
i1
k
pi 1
fŸx1,x2,… ,xk 
n!
x1!x2!Cxk!
p1
x1p2
x2Cpkxk
xi 0,1,… ,n
i 1,2,… ,k
and∑
i1
k
xi n
EŸXi  npi
i 1,2,… ,k
VarŸXi 
npiŸ1 − pi 
i 1,2,… ,k
MŸt1, t2,… , tk−1 
Ÿp1et1 p2et2 C
pk−1etk−1 pk n
ti ∈ g
i 1,2,… ,k − 1
www.
name
-
- ftp.EE.i??p..e.u.,n
Summary of Continuous Distributions
Notation and
Parameters
Probability
Density
Function
fŸx 
Mean
EŸX 
Variance
VarŸX 
Moment
Generating
Function
MŸt 
UniformŸa,b 
b a
1
b−a
a x b
ab
2
Ÿb−a 2
12
ebt−eat
Ÿb−a t t ≠ 0
1 t 0
BetaŸa,b 
a 0, b 0
ßab 
ßa ΓŸb  x
a−1Ÿ1 − x b−1
0 x 1
ß)  ;
0
.
x)−1e−xdx
a
ab
ab
Ÿab1 Ÿab 2
1 ∑
k1
.

i0
k−1
ai
abi
tk
k!
t ∈ g
NŸ6,@2 
6 ∈ g, @2 0
e−Ÿx−6 
2/Ÿ2@2 
2= @
x ∈ g
6 @2 e
6t@2t2/2
t ∈ g
LognormalŸ6,@2 
6 ∈ g, @2 0
e−Ÿlogx−6 
2/Ÿ2@2 
2= @x
x 0
e6@2/2 e
2Ÿ6@2  
−e26@2
DNE
ExponentialŸ2 
2 0
1
2 e
−x/2
x ≥ 0
2 22
1
1−2t
t 12
Two Parameter
ExponentialŸ),* 
) ∈ g, * 0
1
* e
−Ÿx−) /*
x ≥ )
) * *2
e)t
Ÿ1−*t 
t 1*
Double
ExponentialŸ6,* 
6 ∈ g, * 0
1
2* e
−|x−6|/*
x ∈ g
6 2*2
e6t
Ÿ1−*2t2 
|t| 1*
Extreme ValueŸ6,* 
6 ∈ g, * 0
1
* e
¡Ÿx−6 /*−eŸx−6 /*¢
x ∈ g
6 − +*
+ % 0.5772
Euler’s constant
=2*2
6
e6tß1 *t 
t −1/*
GammaŸ),* 
) 0, * 0
x)−1e−x/*
*)ß) 
x 0
)* )*2 Ÿ1 − *t 
−)
t 1*
Inverse GammaŸ),* 
) 0, * 0
x−)−1e−1/Ÿ*x 
*)ß) 
x 0
1
*Ÿ)−1 
) 1
1
*2Ÿ)−1 2Ÿ)−2 
) 2
DNE

欢迎咨询51作业君