and 
for some integers 
and 
Find the value of 
![\[
\int_{0}^{1} \frac{dx}{(\ln x + i\pi)^2\,(1 - x + e^{i\pi})}
\]](http://latex.artofproblemsolving.com/2/7/d/27d389a41148543ea478644e4bc6e0a1258f0d04.png)
and 
for 
Compute ![\[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\]](http://latex.artofproblemsolving.com/1/9/7/197b63d6a2a31e18dae30ef70595b27318667338.png)
in closed form.
,
.
satisfies the quadratic equation 
and deduce that the value of 
.
.
,
is 
.
be the set of all 
matrices with at most two entries in each row equal to 
and all other entries equal to 
.
.
denotes the determinant of the matrix 
be the nearest integer to 
.
Also, generalise your result.
such that 
and 
.
,
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?
