hi AKG,
after thinking it throrughly, I think I understand what you meant.
I only need to solve for Z = R^{-1}SY
and then I show the elements of Z are independent of each other (possibly by showing that their covariance is 0)
Thanks for you help! cheers!
hi thanks for replying
actually, what I need to show is that I can write W in the form RZ where Z are independent.
that is, defining W, S, Y, R, and Z below, and given W = SY where Y are independent, I must show that I can write W as W = RZ where Z are independent.
does what you have...
after a series of computations, I was able to get the following matrix equation from the given of a problem:
\[\left( \begin{array} {ccc} W_1 \\ W_2 \end{array} \right)\] =
\[\left( \begin{array} {ccc} \frac{\sigma_{11}}{\sqrt{\sigma_{11}^2 + \sigma_{12}^2}} &...
Homework Statement
Given a probability space, how do I find the family of all equivalent probabilities on the space?
For example,
let the discrete random variable X have uniform probability equal to 1/3, taking values 10, 2, and -3. the r.v. is represented on the finite probability space...
hi guys,
just submitted the work today...am not sure if what I did was 100% correct, but at least I know it was generally correct.
I did try to show that v_n(1) -> v(1) if v_n conerges to v...something like that :)
just want to say thanks for the help.
the inner product of H1 is given by
(u, v)_{H^{1}} = \int u' v' (x) dx
the norm is
\| \cdot \|_{H^{1}} = \sqrt{(\cdot, \cdot)_{H^{1}}
how exactly do you define a sequence of functions v_n?
thanks.
if my understanding is correct, isn't the l^{2} space a space of measurable functions (and hence the h^{1} space also a space of measruable functions).
My confusion lies is this: I have to find a function that maps the space of measurable function (h1) to the real space. Is this correct?
or is...
okay, let's see if I understood this correctly...
for the first one, I have to find a function that maps from H1 to v(1) (which is equal to 0 and hence is in R).
For the second, I have to find a function that maps from H1 to v(0) - v(1) (which is also equal to 0 and hence is in R).
And since...
thanks for the reply StatusX...
how about for the space V:= \{v \in H^{1}, v(0) = v(1) \}?
How do I prove that this is a Hilbert Space?...more specifically, how do I prove that this is a closed subspace of H^{1}?
thanks again.
Fact 1: we know that a closed subspace of a Hilbert Space is also a Hilbert Space.
Fact 2: we know that the Sobolev Space H^{1} is a Hlbert space.
How do I show that the space V:=\{v \in H^{1}, v(1) = 0\} is a Hilbert space?
Is V automatically a closed subspace of H^{1}? How do I show this...
using the mean value theorem:
Let f(y) = sin y
thenby MVT
f(y) - f(0) = f'(c) (y - 0) where c is between y and 0
getting absolute values of both sides
|f(y) - f(0)| = |y f'(c)|
this is equal to
|sin y - sin0| = |y cos(c)|
equal to
|sin y| = |y cos(c)| <= |y||cos(c)| <= |y| since |cos(c)|...
|sin y| <= |y| for every real y.
I suspect that it is.
It is easy if |y| > 1 since |sin y| <= 1.
but what about for |y| <= 1?
how do I prove this without resorting to the use of graphs?
I need to prove that the function F is Lipschitz, using the
\| \cdot \|_{1} norm.
that is, for
t \in \mathbb{R}
and
y, z \in Y(t) \in \mathbb{R}^{2}
I must show that
\|F(t, y) - F(t, z)\|_{1} < k|y-z|
F(t, Y(t)) is given as
F(t, Y(t)) = \left( \begin{array}{cc} y' \\...