Statistics 310
Lam Ho
Solution for midterm A
1. (25 pts)
(a) (10 pts) The distribution of X is N (0, 1).
(b) (15 pts) Without loss of generality, we can assume = 0 and = 1. So we need
p d
2
to prove that nX
! N (0, 1). It is sufficient to prove that mpnX
Homework 9 Solution
Additional Problems:
1. Problem 1: given the following two independent normal samples with same population variance
12 = 22 = 2
1
Sample 1: Y11 , Y12 , , Y1n1 iid N (1 , 2 )
Sample 2: Y21 , Y22 , , Y2n2 iid N (2 , 2 )
Let
n1
P
Y1 =
n
Homework 7 Solution
1
Additional Problems:
1. Problem 1: given that Y N (0,
2 ),
it follows that
Y
N (0, 1). Recall that square of standard
normal random variable is chi-squared random variable with 1 df, therefore,
Y
2
2 (1).
2. Problem 2:
(a)
FY (y) =
Stat 310
Midterm B
1. Let Y beta(6, 2). Define U = 1
Fall 2013
Y.
(a) Find the pdf of U .
(b) Identify the distribution of U (it is one of the brand name distributions from the back cover of your
book).
2. Suppose that we have two independent samples
Sam
Stat 310
Midterm A
Fall 2013
1. Let X1 , X2 , . be a sequence of iid random variables with mean and variance
2.
Define
n
X
n = 1
X
Xi .
n
i=1
We know the following result. As n ! 1,
p
n
d
n
X
! X.
(a) What is the distribution of X.
(b) Prove this result.
Assignment # 7 Stat 310 Due Friday, 3/10 by 4:00 P.M.
Turn in homework to your TAs mailbox using this sheet as the cover page.
Your TAs Name: Xinyu Song
Fill in your name and also circle the discussion section that you ATTEND. You will pick up the graded
FIN 300 and Your Texas Instruments BA II Plus: Set Up
The BA II Plus makes solving time value of money problems much simpler because it contains most of
the equations, that look fairly daunting, in its memory. To successfully use the BAII you need to set
Homework 3 - Solution
(p.338: 6.75)
Refer to Exercise 6.74. Suppose that the number of minutes that you need to wait for a bus is
uniformly distributed on the interval [0,15]. If you take the bus five times, what is the
probability that your longest wait
Homework 7-Solution
So the CLT can be rewritten as
Y
/ n
~ AN (0,1)
Sn / n p
p (1 p ) / n
S n np
np (1 p )
~ AN (0,1)
~ AN (0,1)
Sn ~ AN (np, np (1 p ).
(c)
Let U n
Sn / n p
n
p (1 p )
( ip1(1 p ) )
n
1
n
Yi np
mZi (t ) E (e Zi t ) E (e
e
Let Wn
tp
p
STAT 310
Midterm 2
Xiran Wang
Midterm 2 - Solutions
1. (a)
E(Sn2 )
1
E
n1
=
1
E
n1
=
( n
)
(Yi Y )
2
i=1
( n
)
Yi2
nY
2
i=1
1
2
(n(2 + 2 ) n(2 + )
n1
n
1
=
(n 1) 2
n1
= 2
=
(b)
1
(Yi Y )2
n1
i=1
( n
)
1
2
Yi2 nY
n1
i=1
)
( n
n
1 2
2
Yi Y
n1 n
n
Sn2
=
=
STAT310
TA : Xinyu(Cindy) Song
[email protected]
STAT 310 Exam I Solution
1. Recall that moment generating function for Y N (, 2 ) is
2 t2
.
mY (t) = exp t +
2
Given that Z =
Y
,
moment generating function for Z is
2
Y
t
t
t
2 (t/)2
Zt
t
Y
= exp
,
Stat 310
Midterm A
Spring 2017
1. (20 points) Let Y N (, 2 ). Derive the distribution of
Z=
Y
.
2. (20 points) Let Y1 , ., Yn be iid geometric(p). Derive the distribution of
Pn
i=1 Yi .
3. (20 points) Let X Beta(, ). Calculate EX and V X.
4. (20 points)
Homework 4 - Solution
(p.394: 8.1)
Using the identity
(
) [ E ( )] [ E ( )
] [ E ( )] B( ) ,
Show that
MSE( )
E[(
) 2 ] V ( ) ( B( ) 2 .
Solution:
Let B
B( ) . Then,
MSE( )
E[( ) 2 ]
E[( E ( ) B) 2 ]
E[( E ( ) 2 ] E ( B 2 ) 2 B E[ E ( )]
V ( ) B 2 .
(p.3
Stat 310
Midterm A
Fall 2014
1. (15 points) Let Y beta(6, 2). Define U = 1 Y . Find the pdf of U .
2. (15 points) Let Y1 , ., Yn be iid geometric(p). Derive the distribution of
Pn
i=1 Yi .
3. (15 points) Let Y1 , ., Yn be iid random variables with EY1 = ,
Stat 310
Midterm B
Fall 2013
1. Let Y beta(6, 2). Define U = 1 Y .
(a) Find the pdf of U .
(b) Identify the distribution of U (it is one of the brand name distributions from the back cover of your
book).
2. Suppose that we have two independent samples
Sa
Stat 310
Midterm B
Fall 2014
1. (30 points) Let Y1 , ., Yn be iid random variables with EY1 = , V (Y1 ) = 2 < . Define the sample
variance to be
n
2
1 X
Sn2 =
Yi Y
n1
i=1
(a) Prove that Sn2 is unbiased.
(b) Prove that Sn2 is a consistent estimator of 2 .
STATS 461
Fall 2016
Homework Assignment #2
Due date: before class on Wednesday, 10/5
1. Normal and Lognormal Distributions.
(a) Suppose that the log return, is normally distributed with mean -0.0124 and standard deviation
0.0675, what is the mean and stan
#
# Donskers Theorem #
#
n=500 # total number of steps in a random walk
t=0.3 # focus on t=0.3, based on the theorem, W_cfw_n,0.3 should converges to
Wiener process W(0.3)
sim=10000 # simulate 10000 random walk paths
W <- numeric(sim)
for ( i in 1:sim)cfw
STATS 461
Fall 2016
Homework Assignment #1
Due date: before class on Friday, 9/16.
Notes:
All tests are based on the 5% significance level.
R packages quantmod and fBasics are helpful in doing this assignment.
1. Suppose that X Bernoulli(p) with p 2 (0,
STATS 461
Fall 2016
Stochastic Process
1. Stochastic Process.
A stochastic process cfw_Xt is a collection cfw_Xt : t I of random variables where the index t
belongs to the index set I . Typically, I is an interval in R (in which case we say that cfw_Xt
Compounding
Money borrowing and lending involve interest. Interest rates
ordinary people work with are compounded annually,
semiannually, quarterly, monthly, weekly or daily.
Example: Suppose that $100 is invested for a year with interest rate 10%
annum.
# Payoff and profit of a call option from a long position at time of maturity
r <- 0.08 # interest rate
premium <- 3 # premium per share
T <- 0.5 # expiration date
K <- 25 # strike price
payoff <- function(x) sapply(x, function(x) max(c(x - K, 0)
profit <
One-step Binomial Tree
Suppose that a stock has price $100/share at current time and its
price at the end of one month will be either $90 (down state) or
$110 (up state). Assume that a call option exists on this stock.
The call option has a strike price o
Association Categorical and Quantitative - Solutions
1: Open the Class Survey (Minitab File) found in the Data Sets folder. Use Stat>Tables>CrossTabulation and Chi-Square to answer the following questions. Put Gender (C2) in the row and the variable Ever