The question I am asking has its reference here. I will list it out simply below. Since I am learning probability properly for the first time, there are a lot of confusion that I want to clarify.
Imagine that we are data scientists tasked with improving the ROI (Return on Investment) of our company's call center, where employees attempt to cold call potential customers and get them to purchase our product. You look at some historical data and find the following:
- The typical call center employee completes on average 50 calls per day.
- The probability of a conversion (purchase) for each call is 4% .
- The average revenue to your company for each conversion is 90 .
- The call center you are analyzing has 100 employees.
- Each employee is paid 200 per day of work.
Before everything, I tried to define the random variables.
Bernoulli
Let $X$ be the success conversion call status of a call made by 1 single randomly chosen employee. In other words, let $X$ be the outcome of a call made by 1 single randomly chosen employee where the outcome is either success or failure in conversion.
Then we can model the probability of success of a call made by a randomly chosen employee as a Bernoulli random variable $X$ with parameter $p$ where $p$ is the probability of success of a call made by that randomly chosen employee.
$\color{red}{\text{Take note how I emphasized the word single, as everything done here is by a single employee.}}$
Binomial
Now the above details a single call made by a randomly chosen employee, which follows a Bernoulli distribution where $X \sim \text{Bernoulli}(0.04)$.
Now the transition from Bernoulli to Binomial should be simple, but I have a big confusion.
A single employee can make multiple calls. For example, in the example above, an employee can make 50 calls a day. We can define a sequence of calls $X_1, X_2, \ldots, X_{50}$. One confusion that arise here is what each $X_i$ mean. In this particular settings, would it be reasonable to assume that each $X_i$ is a copy of $X$ where $X$ is the randomly chosen person. This means once $X$ is determined (tagged to the person chosen, say person A), then $X_1, X_2, \ldots, X_{50}$ all refer to this person A's calls as it does not make sense that $X_i$ corresponds to a different person.
In any case, since $n=50$ and $p=0.04$, we can define our $Y$ as follows:
$$ Y \sim \text{Binomial}(50, 0.04) $$
where $Y = X_1 + X_2 + \ldots + X_{50}$ since each call made by a random employee is assumed to be i.i.d.
Now the second confusion (assuming there is no conceptual error before this, but I doubt it) is the interpretation of $Y_i$.
In this context, can we be unambiguous and say that each $Y_i$ should be the number of successful conversion calls made by a single randomly selected employee.
To be more precise, each experiment for $Y_i$ is like a new randomly drawn 50 calls by a random person, so each time is a realisation of 50 calls from a randomly selected person. So if you have 100 people you can have 100 $Y_i$. Note the distinction, in $X_i$, this $X_i$ is bind to a single person (may be wrong here), here each $Y_i$ can be any member from the population.
I understand this is a long question, but would like to seek some validation on whether my understanding above is outright wrong, or somewhat close. I am taking the course alone and won't be able to ask others to clarify conceptual understanding.
