I have a elevator probability problem but with a twist (instead of people exiting the elevator, it is the elevator stopping on each level). Require some help understanding and completing the probability questions.

**Problem:**

4 people go into the elevator of a 5-storey shophouse. Assume that each of them exits the building uniformly at random at any of the 5 levels and independently of each other. N is the random variable which is the total number of elevator stops.

1. Describe the sample space for this random process.

**Answer:** {1,2,3,4,5}

**Understanding:** Sample space represents all the possible outcomes of the random event. Hence, the elevator can only stop at these 5 floors. Just like throwing a die and randomly getting a number, the sample space for that is {1,2,3,4,5,6}.

2. Let $X\_i$ be the random variable that equals to 1 if the elevator stops at floor i and 0, otherwise. Find the probability that the elevator stops at both floors $a$ and $b$ for $a$, $b$ $\in$ {1,2,3,4,5}. Find $EX\_aX\_b$.

**Question:** Do they mean the elevator stop at two consecutive floors or two different floors? Can a and b be the same, a smaller than b and vice versa?

3. Prove the independence of $EX\_1X\_2$.
4. Calculate $EN^2$ (represented as ($X\_1 + ...+ X\_5)^2 = \sum\_{(i,j)} X\_aX\_b$ where the sum is over all ordered pairs (a,b) of numbers from {1,2,3,4,5} and the linearity of expectation. Find the variance of N.
5. Determine the distribution of N. That is, determine the probabilities of events N = i. Compute $EN and EN^2$ directly by using the laws of expectation.

I got stuck from question 2 onwards with thinking of the possible ways that this can be done. It would be superb if someone can shed some light on them. Many thanks.