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This page lists two projects related to things we have learned. Do any one of them. Each is worth 10 marks.
Whichever project you do, you need to submit a report (handwritten/pdf) along with the code (if any).
In one of the exercises about $Unif(0,1)$ distribution, we had partitioned $(0,1)$ into a number of equal subintervals,
and had permuted them. This "scrambling" had produced a new random variable from an $Unif(0,1)$ random variable. Your
first task is to redo that exercise to convince yourself that the new random variable again has $Unif(0,1)$ distribution.
Now take any $X\sim Unif(0,1),$ and split $(0,1)$ into $n$ equal subintervals
where $n\in{\mathbb N}.$ Call these subintervals the states, labelled $1,...,n.$ Now
split each state into $n$ further subintervals (equal width) called districts, again
lablled $1,...,n.$ Thus, effectively you have split $(0,1)$ into $n^2$
subintervals that may be labelled as $(i,j)$ for $i,j\in\{1,...,n\},$ where $i$ denotes the state, and
$j$ denotes the district within the state.
Let $Y$ be obtained by scrambling $X$ as follows: Swap $(i,j)$-th subinterval with the $(j,i)$-th subinterval.
Your next task is to find the joint distribution of $(X,Y).$ For this draw the joint distribution of $(X,X)$
as the $45^\circ$ line drawn as a diagonal in the unit square. The entire mass 1 is uniformly distributed along this
line. Take $n=2$ and do the swaps in the vertical axis, and sketch how the diagonal line
gets split into $n^2=4$ equal pieces and gets redistributed.
Obtain the joint CDF of $(X,Y)$ as the points $\left(\frac in,\frac jn\right).$
Compare with the CDF had $X, Y$ been IID $Unif(0,1).$
Do you notice anything funny?
For this project you need to know how to extract pixel values from an image. There are R and
Python packages/functions for this purpose. You may Google, or ask me to learn about them.
Take printed text, preferrably large font, black ink on white paper. Hold your camera fixed in
from of it, and take 50 greylevel photos of the same thing without moving th camera or the page. Many
cameras have a "burst mode" that will help you to take a bunch of photos with a single click.
Extract the pixel values for each. Compute $V_{ij}=$ variance of the 50 meaurements of the
$(i,j)$-th pixel. Would you expect all the $V_{ij}$'s to be nearly the same? Think, and make a guess. Now actually
compute and see.
Can you explain what you see?
Let $k\in{\mathbb N}$ be given.
For $\alpha_1,...,\alpha_k\in(0,1)$ with $\sum \alpha_i=1$ consider the mixture
distribution with CDF given by $F(x)=\sum \alpha_i \Phi\left(\frac{x- \mu_i}{\sigma_i}\right)$
for some $\mu_i$'s and $\sigma_i$'s.
Show that given the function $F(x)$ one can, in principle, find $\alpha_i$'s and
$\sigma_i$'s uniquely. In other words, the map $(\alpha_1,...,\mu_1,..., \sigma_1,...)\mapsto F$ is one-one.
Working with $k=2$ would be enough.