Let be a standard Brown motion.
a) Show that
b) What is the correlation of X(t) and X(s) ?
Let be a standard Brown motion. a) Show that b) What is the correlation of X(t) and X(s) ?
Let X(t) = μt + σB(t) be a standard Brown motion Show that: Cov(X(s), X(t)) = σ2 min(s, t)
Let be iid observations from , is known and is an unknown real number. Let be the parameter of interest. (a) Find the CRLB for the variance of an unbiased estimator for . (b) Find the UMVUE for . (c) Show that is an unbiased estimator for . (d) Show that . We were unable to transcribe this imageσ2 (μ, ) We were unable to transcribe this imageWe were unable to transcribe this imageg(t) = 211 We were unable to...
Show that the correlation matrix of any random vector X is nonnegative definite, where the correlation matrix is defined by , (Assume we know that the covariance matrix of X denoted is defined by is nonnegative definite, and . Re IRmxm We were unable to transcribe this imageVar(Xi)Var(X ат ат We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
a) Let
. Show that
.
b) Show that the derivative can be written as:
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consider the variation of constants formula where P(t)= a) show that solves the initial value problem x'+p(t)=(t) x()= when p and q are continuous functions of t on an interval I and tg p(s)ds We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image tg p(s)ds
Let
and define
by .
(a) Show is one-to-one
(b) What is the formula for
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Let
be an arbitrary function and A
X.
i) Show that A
ii) Give an example to show that in general A =
.
iii) Show that, if
is injective, then A =
iv) Show that, if X and Y are modules;
is a homomorphism of modules and A is a submodule of X such that
ker,
then we also have A =
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Please show all work:
Let
If x is odd then
If x is even then
Prove that
is true and then solve it.
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Let X be a banach space such that X= C([a,b]) where - ab+ with the sup
norm. Let x and f X. Show
that the non linear integral equation
u(x) = (sin
u(y) dy + f(x) ) has a solution u X. (the integral is
from a to b).
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Let X(t) =
2; if 0 t 1;
3; if 1 t 3;
-5; if 3 t 4:
or in one formula X(t) = 2I[0;1](t) +
3I(1;3](t) -
5I(3;4](t).
Give the Itˆo integral
X(t)dB(t)
as a sum of random variables, give its distribution,
specify the mean and the variance.
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