Question

let (X,S,U) be a measure space.show that U is a monotone

Answer #1

Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v; w); z = z(u;
v; w). To calculate ∂s ∂u (u = 1, v = 2, w = 3), which of the
following pieces of information do you not need?
I. f(1, 2, 3) = 5
II. f(7, 8, 9) = 6
III. x(1, 2, 3) = 7
IV. y(1, 2, 3) = 8
V. z(1, 2, 3) = 9
VI. fx(1, 2, 3)...

1)
a) Let z=x4 +x2y, x=s+2t−u, y=stu2:
Find:
( I ) ∂z ∂s
( ii ) ∂z ∂t
( iii ) ∂z ∂u
when s = 4, t = 2 and u = 1
1) b> Let ⃗v = 〈3, 4〉 and w⃗ = 〈5, −12〉. Find a
vector (there’s more than one!) that bisects the angle between ⃗v
and w⃗.

Let S, U, and W be subspaces of a vector space V, where U ⊆ W.
Show that U + (W ∩ S) = W ∩ (U + S)

Let X U (0, 1) and Y exp (1) be independent variables (v.a.’s)
independent. What is the function probability density (f.d.p.) of
the v.a. Z = X + Y?

Solve the differential equation:
x*(du(x)/dx)=[x-u(x)]^2+u(x)
Hint: Let y=x-u(x)
Answer: u(x)=x-x/(x-K) with K=constant

Let U = Z+ ∪ { 0 }.
Let R1 = { ( x, y ) | x ≠ y }
Let R2 = { ( x, y ) | x = y }
Let R3 = { ( x, y ) | x ≥ y }
Let R4 = { ( x, y ) | x ≤ y }
Let R5 = { ( x, y ) | x > y }
Let R6 = { ( x, y...

Let U and V be two independent standard normal random variables,
and let X = |U| and Y = |V|.
Let R = Y/X and D = Y-X.
(1) Find the joint density of (X,R) and that of (X,D).
(2) Find the conditional density of X given R and of X given
D.
(3) Find the expectation of X given R and of X given D.
(4) Find, in particular, the expectation of X given R = 1 and of...

Q8) Let u(x) = 2x2 and
f (x) = [u(x)]3 =
8x6. Which of the following is f
´(x)?
(a) 3[u(x)]2 (b)
3[u´(x)]2 (c)
3[u(x)]2u´(x)
Q9) A rock is thrown into a still pond and causes a circular
ripple. If the radius of the ripple is increasing at 2 feet per
second, how fast is the area changing when the radius is 10 feet?
[Use A = πR2]

An observer in frame S′ is moving to the right (+
x-direction) at speed
u = 0.650 c away from a stationary observer in
frame S. The observer in S′ measures the speed
v′ of a particle moving to the right away from her.
a)What speed v does the observer in S measure for
the particle if v′= 0.430 c ?
b)What speed v does the observer in S measure for
the particle if v′= 0.870 c ?
c)What speed...

Let f be monotone increasing and absolutely continuous on [0,
1]. Let E be a subset of [0, 1] with m∗(E) = 0. Show
that m∗(f(E)) = 0. Hint: cover E with countably many
intervals of small total length and consider what f does to those
intervals. Use Vitali Covering Argument

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