$5\mathrm{.}2[2pt]$ Suppose that there is a $3-$regular graph $($i$.$e$.$ each vertex is connected with an edge to three other vertices$)$ with $n$ vertices in total. To each vertex, we assign a random number from a uniform distribution over $0,1.$ Then we select a set of vertices using the following rule: a vertex will be selected if its random number is greater than the random numbers of all of its three neighbors. Write a function called vertices$\_$selected that takes the number of vertices, $n,$ as an input and returns the expected number of vertices that will be selected. For this question, the input $n$ will always be such that a $3-$regular graph is possible.

Hint: Use Trick #$1$ from the lecture about expectation, and let $x_i$ be the random variable that equals $1$ if vertex $i$ is selected and equals $0$ if it is not selected.

In $[]$: def vertices$\_$selected$($n$)$:

# YOUR CODE HERE

raise NotImplementedError$()$

In $[]$: #hidden tests for problem $5\mathrm{.}2$ are within this cell

$5\mathrm{.}3[1pt]$ Write a function called uni$\_$variance that takes $b$ and $c$ as inputs and returns the variance of a uniform random variable over $b,c.$ Hint: Use $($i$)$ the result about the variance of a uniform r$.$v$.$ over $0,1,$ and $($ii$)$ Trick #$2$ from the lecture about variance.

In $[]$: def uni$\_$variance $(b,c)$ :

# YOUR CODE HERE

raise NotImplementedError$()$

In $[]$: #hidden tests for problem $5\mathrm{.}3$ are within this cell