A set X is said to be closed under multiplication if for every
x1,x2 ∈ X...
A set X is said to be closed under multiplication if for every
x1,x2 ∈ X we have x1x2 ∈ X. Let A be the union of all bounded
subsets X ⊆ R that are closed under multiplication. Does inf(A)
exist? If it does, find it.
The set R^2 with addition and scalar multiplication defined
by
(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
c(x1, y1) = (cx1, y1)
is not a vector space. Determine which axiom fails and find a
counterexample that shows that it fails.
Does the input requirement set
V (y) = {(x1, x2, x3) | x1 + min {x2, x3} ≥ 3y, xi ≥ 0
∀ i = 1, 2, 3}
corresponds to a regular (closed and non-empty) input
requirement set?
Does the technology satisfies free disposal? Is the technology
convex?
Q. A sequence X1, X2, ... , Xn is said to be cyclically sorted
if the smallest number in the sequence is Xi for some unknown ?,
and the sequence Xi, Xi+1, Xn, ... , X1, X2,....Xi-1 is sorted in
an increasing order. Design an algorithm to find the position of
the minimal element in a cyclically sorted ? distinct elements. (6
points) If your algorithm uses recursion, you need so show the
recurrence function. Otherwise, show a closed-end form...
Using Matlab
1. Solve the following equations set
f1 (x1,x2) = sin (sin (x1)) +x2
f2 (x1,x2) = x1+ e^(x2)
a) Can this equation set be solved by the fixed - point
method with the following expressions? And why? Show your analysis
with a 2D graph.
g1 (x1,x2) = -e^(x2)
g2 (x1,x2) = -sin(x1)
b) Use Newton Raphson Method with initial values x1 =
-2, x2 = 1.5. (8 significant figures. Please submit the code and
results.)
If the joint probability distribution of X1 and X2 is given by:
f(X1, X2) = (X1*X2)/36 for X1 = 1, 2, 3 and X2 = 1, 2, 3, find the
joint probability distribution of X1*X2 and the joint probability
distribution of X1/X2.
2.2.8. Suppose X1 and X2 have the joint pdf
f(x1, x2) = "
e−x1 e−x2
x1 > 0, x2
> 0
0 elsewhere
.
For constants w1 > 0 and w2 > 0, let W = w1X1 + w2X2.
(a) Show that the pdf of W is
fW (w) = "
1
w1−
w2
(e−w/w1 − e−w/w2) w > 0
0 elsewhere
.
(b) Verify that fW (w) > 0 for w > 0.
(c) Note that the pdf fW...
Bridgit’s utility function is U(x1, x2)= x1 + ln x2 x1 -
stamps
x2 - beer
Bridgit’s budget p1 x1 + p2 x2 = m
p1 – price of stamps
p2 – price of beer
m – Bridgit’s budget
a) What is Bridgit’s demand for beer and stamps?
b) Is it true that Bridgit would spend every dollar in
additional income on stamps?
c) What happens to demand when Bridgit’s income changes (i.e.
find the income elasticity)?
d) What happens...
Answer each of the following statements True/False/Uncertain.
Give a full explanation of your answer including graphs where
appropriate. (When in doubt, always include a fully labeled
graph.)A) Consider the following three consumption bundles
(X1,X2)=(10,10) ; (X1,X2)=(15,10) ; (X1,X2)=(3000,8). Non-satiation
implies that (15,10) is preferred to (10,10) but does not imply
that (3000,8) is preferred to (10,10).B) It is not theoretically possible for two indifference curves
to cross if the preference relations they are based on satisfy the
assumptions of completeness,...
Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ]
= (µ1, µ2) and var[X] = σ11 σ12 σ12 σ22 . (a matrix)
(i) Let Y = a + bX1 + cX2. Obtain an expression for the mean and
variance of Y .
(ii) Let Y = a + BX where
a' = (a1, a2) B = b11 b12 0 b22 (a matrix).
Obtain an expression for the mean and variance of Y .
(ii)...