In a game of fantasy football players score points each week depending on the team they have selected, and at the end of a season the player with the most points wins. There are $W = 38$ weeks, and a total of $N$ players. In the $w$th week I would like to estimate the probability that the $i$th player will win the league, given his or her recent form.
As a rough approximation, I suspect that each player's weekly score will follow a normal distribution, with a player specific mean, $\mu_{i}$, and a player specific standard deviation, $\sigma_{i}$. I can calculate these statistics for any given week by looking at player data from a pre-determined number of previous weeks.
If a player's score in week $w$ is $s_{iw}$, and there are $W - w$ games left, their expected final score, $s_{iW}$, is simply given by $$s_{iW} = s_{iw} + \mu_{i} (W - w),$$ which is just their current score plus any additional points they are expected to gain in the remaining weeks. I'm not sure, however, how to calculate the standard deviation around the final score. More importantly, the player with the highest score, $s_{iW}$, is obviously the most likely to win, but how do I calculate the probability that they will win? I will essentially have $N$ overlapping normal distributions for the final scores. Do I need to sum up the pairwise probability that each player beats another or something like that?
This is just for a bit of 'fun', and a good way to get me to learn some statistics, so any advice is much appreciated. References to introductory materials about these kinds of questions also welcomed.