3.4- Let Y1 = θ0 + ε1 and then for t > 1 define Yt recursively...
3.4- Let Y1 = θ0 + ε1 and then for t > 1 define Yt recursively
by Yt = θ0 + Yt−1 + εt. Here θ0 is a constant. The process {Yt} is
called a random walk with drift.
Let {an} be a sequence defined recursively by a1 = 1 and an+1 =
2√ 1 + an where n ∈ N
(b) Does {an} converge or diverge? Justify your answer, making
sure to cite appropriate hypotheses/theorem(s) used. Hint : Try
BMCT [WHY?].
(c) (Challenge) If {an} converges then find its limit. Make sure
to fully justify your answer.
23-64) Let Yt be the sales during
month t (in thousands of dollars) for a photography
studio, and let Pt be the price charged for portraits
during month t. The data are in the file Week 4 Assignment
Chapter 12 Problem 64. Use regression to fit the following model to
these data:
Yt = a + b1Yt−1 + b2Pt + et
This equation indicates that last month’s sales and the current
month’s price are explanatory variables. The last term, et,...
1. . Let X1, . . . , Xn, Y1, . . . , Yn be mutually independent
random variables, and Z = 1 n Pn i=1 XiYi . Suppose for each i ∈
{1, . . . , n}, Xi ∼ Bernoulli(p), Yi ∼ Binomial(n, p). What is
Var[Z]?
2. There is a fair coin and a biased coin that flips heads with
probability 1/4. You randomly pick one of the coins and flip it
until you get a...
Question 1
A) Show that the functions y1(t) = 1 + t 2 ; y2(t) = 1 − t 2 are
linearly independent directly from the definition of linear
independence.
B)Find three functions y1(t), y2(t), y3(t) such that any two of
them are linearly independent but three of them are not linearly
independent.
Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1
0, otherwise. a) Are Y1 and Y2 independent? Why? b) Find Cov(Y1,
Y2). c) Find V(Y1−Y2). d) Find Var(Y1|Y2=y2).
The parametric equations
x = x1 +
(x2 −
x1)t, y
= y1 +
(y2 −
y1)t
where
0 ≤ t ≤ 1
describe the line segment that joins the points
P1(x1,
y1)
and
P2(x2,
y2).
Use a graphing device to draw the triangle with vertices
A(1, 1), B(4, 3), C(1, 6). Find the
parametrization, including endpoints, and sketch to check. (Enter
your answers as a comma-separated list of equations. Let x
and y be in terms of t.)
The accompanying data file contains 20 observations for
t and yt.
t
1
2
3
4
5
6
7
8
9
10
yt
13.7
8.9
10.4
9.3
12.1
13.7
11.9
13.9
13.8
13.5
t
11
12
13
14
15
16
17
18
19
20
yt
10.8
10.6
12.3
9.3
11.2
10
10.5
13.7
10.3
10.7
The data are plotted below.
a. Discuss the presence of random variations.
A. The smoother appearance of the graph suggests the presence of
random variations....
The accompanying data file contains 20 observations for
t and yt.
t
y
1
10.32
2
12.25
3
12.31
4
13
5
13.15
6
13.84
7
14.39
8
14.4
9
15.05
10
14.99
11
16.95
12
16.18
13
17.22
14
16.71
15
16.64
16
16.26
17
16.77
18
17.1
19
16.91
20
16.79
b-1. Estimate a linear trend model and a
quadratic trend model. (Negative values should be indicated
by a minus sign. Round your answers to 2 decimal
places.)...
Let c(y1, y2) = y1 + y2 + (y1y2)^ −(1/3). Does this cost
function have economies of scale for y1? What about economies of
scope for any strictly positive y1 and y2. Hint, economies of scope
exist if for a positive set of y1 and y2, c(y1, y2) < c(y1, 0) +
c(0, y2). [Hint: Be very careful to handle the case of y2 = 0
separately.]