Write the code that will compute the log likelihood over the entire dataset. Report the log likelihood for each of the three considered estimates

of theta $0\mathrm{.}5,1\mathrm{.}0,2\mathrm{.}2.$ Which of the estimates gives the highest log$-$likelihood?

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# Please write your code below

theta$\_$hats $=[0\mathrm{.}5,1\mathrm{.}0,2\mathrm{.}2]$

Answer:

Question $5$:

$($Finding the Maximum Likelihood Estimator$)$

We want to find the parameter value hat$()$ that makes our data set most likely:

hat$({}_{\text{M}\text{L}\text{E}})=argma{x}_{}L(|x,Y)$

As shown in lecture, we know that hat$()$ for the MLE $($and the one that minimizes MSE$)$ is:

hat$({}_{\text{M}\text{L}\text{E}})=$hat$({}_{\text{M}\text{S}\text{E}})=\frac{{\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i{y}_i}{{\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i^2}$

Compute and report theta$\_$hat$\_$MLE using the formula above

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# Compute theta$\_$hat$\_$MLE

Plot the log likelihood for thetas at a range of values between $0\mathrm{.}1$ and $2\mathrm{.}0.$ Does the peak of this curve match the value calculated above?

### generate plot of the curve below and you can estimate the peak by eye