[/quote]
,
such that there exists a prime number 
,
is a power of 
.
where 
is a positive integer.
atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon 
to any other usamon 
.
,
.
of 
at point 
,
meet 
.
is a tangent of 
.
,
.
meet 
,
is on the line segment 
meet 
.
such that 
.
be a function such that 
.
.
be positive integers and 
and 
.
is a perfect square.
residues 
.
contains at least half of all the residues 
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
be a twice differentiable function such that![\[
f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1.
\]](http://latex.artofproblemsolving.com/1/e/1/1e178debf8ae0b6b76d2e60ffca95f5a4c4ad4e5.png)
Prove that there exists a point 
such that![\[
f(\xi) \cdot f'(\xi) + f''(\xi) = 0.
\]](http://latex.artofproblemsolving.com/4/6/b/46b8d103981b838ae0aa5afdee104edfae4e4b8f.png)
and there is a morphism 
to the monoid 
such that for all 
,
such that 
.
positive such that 
,
positive such that 
converges and 
diverges ?
is continuous, positive with 
,
such that 
converges and 
diverges ?
is in the 
class, increasing : I chose 
.
but it is not clear that it is ok, even if 
,
diverges ?
converges ?
does not converge to 
,
,
Y by
Suppose we are given i.i.d.\ observations 
from a distribution with probability density function (PDF) 
for 
, where the parameter 
has a prior distribution with PDF 
. Consider the following two approaches to Bayesian updating:
(1) Let
be the complete data vector. Denote the posterior PDF as 
, where 
, obtained by applying Bayes' rule to the full dataset at once.
(2) Start with prior
. For each , let 
be the current prior and update it using observation 
to obtain the new posterior:
Are the final posteriors from part (a) and 
from part (b) the same? Provide a proof or a counterexample.
Here is the proof I've written:
Do you guys think this is rigorous enough? What would you change?
from a distribution with probability density function (PDF) 
for 
, where the parameter 
has a prior distribution with PDF 
. Consider the following two approaches to Bayesian updating:(1) Let
be the complete data vector. Denote the posterior PDF as 
, where 
, obtained by applying Bayes' rule to the full dataset at once.(2) Start with prior
. For each , let 
be the current prior and update it using observation 
to obtain the new posterior:
Are the final posteriors
from part (a) and 
from part (b) the same? Provide a proof or a counterexample.Here is the proof I've written:
Do you guys think this is rigorous enough? What would you change?
