Part $2$: A Gaussian model of reading

Suppose you do a reading experiment and collect reading times data $y.$ The data $y$ contains $n$ in$-$

dependent and identically distributed datapoints $y_1,y_2,$dots,$y_n.$ You make the following assumptions

about the underlying generative process:

$($a$)$ The likelihood assumption

Each datapoint $y_i$ comes from a normal distribution with mean $$ and standard deviation $$ :

$y_{i}Normal(,)$

The joint likelihood of the data $y_1,y_2,$dots,$y_n$ can be given by:

$L(,|y)=\frac{1}{(\sqrt[\phantom{2}]{2}{)}^n}{e}^{-\frac{1}{2{}^2}{\displaystyle \underset{i=1}{\overset{n}{}}}(y_i-{)}^2}$

$($b$)$ The prior assumptions

Based on the previous research on the topic, you make the following assumptions about the stan$-$

dard deviation $$ and the mean reading times $.$

$=50$

$Normal(250,25)$

The product of the likelihood and the prior density gives you unnormalized posterior density.

The unnormalized posterior density of $$ is given by

$p^{\text{'}}(|,y)=L(,|y)p()$

Suppose the observed reading times data $y$ consists of $8$ observations: $300,270,390,450,500,290,$

$680,450.$

Answer the following:

$2\mathrm{.}1$ Given the above data $y$ and the likelihood and prior assumptions, calculate the unnormalized

posterior density for the following values of $.$

$=300$

$=900$

$=50$

$2\mathrm{.}2$ Graph the unnormalized posterior distribution of $.$

$2\mathrm{.}3$ Compare the graphs of the prior and the $($unnormalized$)$ posterior distribution of $.$

From assignment $2$ onwards, we will get coding exercises,you can submit using a Python notebook but a PDF file. Please give solution in python.