An update to my previous post regarding season length and Pythagorean records:
I received several inquiries (zero) as to how I performed the integration discussed in my last post. I coded a very rough numerical integration routine to perform the integrations. I chose this route because it was the quickest and easiest way to generate results. The problem with the numerical integration is that you, dear reader, cannot fiddle with different situations that interest you. An analytic solution for the integral can be obtained, however, and I now present to you the result:
where P(y>x) is the probability that a team with Pythagorean winning percentage y_bar will match or outperform a team with Pythagorean winning percentage x_bar over N games (x_bar should be greater than y_bar). Many of you may not be familiar with the function erfc. erfc is the "complementary error function," a tabulated function that appears often in statistics and engineering. If you want to know the value of erfc, you will need either a table (available online) or software such as Excel that can calculate the error function. [Aside: syntax for usage in Excel is "=erfc(arg)", where arg is the argument of the complementary error function.] Recall the limitations of this analysis from last time: the teams play independent games and runs scored and runs allowed are roughly independent.
Anyway, the point of all this mathematics is to allow you to play around with different scenarios. Say you want to know how often a .600 Pythagorean team is outplayed by a .400 Pythagorean team over a 30-game stretch. Just use the formula as prescribed:
Voila! The probability that a .600 team is outplayed by a .400 team over a 30-game stretch is 6.1%. Not likely, but not as uncommon as I might have guessed.
Finally, I'd like to point out that the analytic integration was not performed by yours truly - I'm not nearly clever enough. My fiancee is a very talented physicist whose mathematical skill is unparalleled in our househould. She has been enormously tolerant, often encouraging, and occasionally enthusiastic about my spending hours in front of the computer obsessing over baseball minutae. For her to participate in my sabermetric hobby is above and beyond the call of wifely duty. Lou Gehrig was wrong: it is I, not he, who is the luckiest man on the face of the earth.