Information. You are planning to run a computationally intensive data science task on one of your favorite cloud service providers' compute nodes. The cloud provider has access to many nodes of different compute capabilities, and will assign your task to one of them, but you don't know what node that will be$.$

Say that each compute node is characterized by a "speed index" $x$ that is distributed as a uniform distribution in $1,2.$ Given that one node's speed index is $x,$ the run time for the task you want to run on is given by an exponential random variable with parameter $=x.$ Larger index implies faster execution, typically. $($Hint: remember that the conditional PDF you will need to calculate the MLE is nonzero only when $x$ is in $1,2)$

The figure below illustrates the conditional PDF of $Y$ for various values of $x.$

After you run your task, and observe its runtime $Y,$ you would like to estimate the speed index of the node you were assigned. In this exercise we will try a few different estimators for $x,$ i$.$e$.,$ the speed index, given that we observed the runtime $Y.$