Problem Set II:

On the web site you find a data set from a hypothetical instrument: 
Each row of the file is one 5x5 pixel image formatted like this:

c01 c02 c03 c04 c05
c06 c07 c08 c09 c10
c11 c12 c13 c14 c15
c16 c17 c18 c19 c20
c21 c22 c23 c24 c25

(where c1 means the first column, column 13 is the center pixel of the image)

It has imaged a source known to be at the center of the image.

The noise in each each image is statistically independent from previous 
images, but may be correlated between pixels.

Assume the instrument has a gaussian psf and that the source's width
is much smaller (i.e., the source is unresolved).

1) Find the pixel-pixel covariance matrix for the data set.

2) Plot the 2d likelihood for source amplitude and psf width (sigma).  

3) Plot the 1d likelihood for source amplitude, marginalized over psf width.  
From this find the 68% and 95% confidence intervals, assuming a uniform prior 
on source amplitude.

4) For this type of source, the integral source counts on a logN vs logS plot 
has a slope of -2.5. (N is the number of sources brighter than a given flux 
density (S)).  Re-do (3) plotting P_{A|D} instead of the likelihood (P_{D|A} 
with the prior inferred from this. (A is amplitude, D is data)