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)
positive such that 
,
positive such that 
converges and 
diverges ?
is continuous, positive with 
,
continuous and positive exist on 
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 
,
has always integral solutions for 
and 
whenever 
is a positive integer.
and 
.
.
be a function such that 
and for 
Prove that
exists and is less than 
,
.
and 
![$A = \sum_{r=0}^{24} \frac{d(F(r,x))}{dx} \left[\frac{1}{F(r,x) + \frac{1}{F(r,x)}}\right]$](http://latex.artofproblemsolving.com/4/2/b/42bd0116f69f03884746ba8d06493ed999a567d6.png)
![$B = \sum_{r=0}^{24} \frac{-d(G(r,x))}{dx} \left[\frac{1}{G(r,x) + \frac{1}{G(r,x)}}\right]$](http://latex.artofproblemsolving.com/9/f/5/9f5ff4b5d991249940fa0521b222e5e4201a9fe0.png)
![$R = \frac{A - B}{\tan x\left[PQ - \frac{1}{3^{25}}\right]}$](http://latex.artofproblemsolving.com/b/f/4/bf447429d8f9e71117178abb46c688d7d4a036d6.png)
where 
is divided by 
.
,
satisfied
for all 
and all 
.
.
such that 
is uniformly continuous on 
.
be a closed map with the property that the inverse image of each point in 
is a compact subset of 
.
is compact whenever 
is compact in 
,
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?
