$1$ Probability Mass Functions $(30$ points$)$

As a sales representative for a CAEE firm, your mission is to market top$-$notch construction equipment

to office workers within a bustling business district. This district boasts a total of six CAEE organizations,

consisting of three major corporations and three small, local businesses, with their locations and

connecting routes outlined as depicted. Your product is of NO interest to these tech$-$oriented organizations

and they will say no to you regardless of how much you persuade $($you know this apriori, but alas, your

company is trying all possible ways to sell the product$).$ Being a salesman with a persistent attitude, you

adopt the following strategy:

you pick an organization at random and ask them if they want to buy your product.

If they say no$,$ you pick one of the two neighboring organizations with probabilities $0\mathrm{.}4$ and

continue the process OR you give up and decide to return back with probability $0\mathrm{.}2.$

You keep repeating this process until you give up and decide to return back to your

supervisor.

Let $x$ be the total number of times you ask any organization to purchase your product.

$(2$ points$)$ What is the range of variable $x?$

$(6$ points$)$ Derive the general formula for probability mass function of $x?$

$(4$ points$)$ Verify and show that ${\displaystyle \underset{x}{\overset{?}{}}}p(x)=1.$

$(6$ points$)$ Does $x$ have a negative$-$binomial distribution? If yes, find the values of $m$ and $p$ and

use those to derive the expected value of $x.$

For this same problem, Let $Z$ be the number of times you ask a big organization to purchase

your product.

$(5$ points$)$ Write down all possible outcomes in which $Z$ is equal to $2.$

$(7$ points$)$ Derive the probability that $Z=2.$