Testing for Normality

• The mean(5.7) and the median (5.2) show that our distribution isn’t “normal”. However, we can test our data to see what percentage of it’s distribution falls within 2 \(\sigma\) to get a practical sense of our distribution. As seen below 94% of our distribution falls within 2 \(\sigma\) , I would say that for all practical purposes, the win rate is normally distributed as we would expect thanks to the central limit theorem.

## Estimated average bb/100
average_win_rate <- mean(win_rate_array_df$win_rate_array)

## build out sd test for normality check
upper_end <-  sd(win_rate_array_df$win_rate_array) * 1.96 + average_win_rate
lower_end <-  sd(win_rate_array_df$win_rate_array) * -1.96 + average_win_rate

## grab length of entire distribution
total_count <-  length(win_rate_array_df$win_rate_array)

## find length of samples within 2 sd
samples_within_2sd <- win_rate_array_df %>% 
  filter( win_rate_array >= lower_end & win_rate_array <= upper_end) %>% 
  summarise(length(win_rate_array))


## find percent of samples within 2 sd
print(samples_within_2sd / total_count)

##   length(win_rate_array)
## 1              0.9406088

Dirty Data

Total Aggregated Statistics By Player

The results above are the aggregated player level statistics. Attempting to run a regression model on this data was failing. A large reason for this is the massive variance you saw in the Vegas example. To compound the issue, players with 55 hands needed to be treated in a different way than players with 50k hands, as win rate only approaches the true mean win rate over a large sample. So my models were learning on garbage data. My solution was to build out custom confidence intervals for win rates, and discard players which had an interval spanning 0. However, my stats aren’t distributions and therefore I have no samples and no \(\sigma\) . Having hands which could be treated as samples I needed to figure out a good way to estimate \(\sigma\) . I looked at player playing styles, more aggressive players I assumed had a higher \(\sigma\) . Something like this below

• Create some intervals. VPIP is how many hands a player plays out of total hands they were dealt. High VPIP likely leads to high \(\sigma\)

• vpip<10 will be assigned a \(\sigma\) of 60
• 10<vpip<20 will be assigned a \(\sigma\) of 70
• 20<vpip<30 will be assigned a \(\sigma\) of 80
• 30<vpip<40 will be assigned a \(\sigma\) of 90

• 40<vpip<100 will be assigned a range of 110

I then labeled winning players and losing players and ran a logistic regression to classify a player as either winning or losing. My hope was that eliminating some of the noise would lead to a better model. I haven’t attempted to rerun, tune, or evaluate the classification with other metrics, but overall it correctly identified 13 out of 17 winners and 64 out of 65 losers on a test set