It is common in continuous distributions that the average result is more probable than the extreme results. The triangle distribution satisfies this condition.
The checks on the equations of the uniform distributions aren't particularly satisfactory, insofar as they really only confirm that the leading term is correct. For a more detailed test, we calculate a continuous form of our earlier discrete two class problem, to see if we get the same answer.
The first distribution to be considered is the Uniform distribution. In a uniform distribution, there is a continuous set of possible values, each of which are equally likely.
There's no particular reason to assume that the distribution of probabilities must come in discrete lumps. So we now analyze continuous distributions - those in which the probability distribution is smeared over a number of possible values.
Before jumping into the three distributions which will be covered, it is worth taking a moment to preview the work that is to come.
The next exercise is to generalize our previous result. We continue to maintain the overall average, and examine what happens when we vary the magnitude of the bias.
In other words, instead of bags of coins that land tails with probability .5 + .25 and .5 - .25, we work through the same exercise with bags of coins that land tails with probability .5 + x and .5 - x. For my own sanity, let us define the median value M = .5.
We continue with our bags of biased coins, this time considering two different questions. Having flipped the coins, and collected 82 that all landed tails each time, we ask
What is the probability that a randomly chosen coin in this group is biased against tails?
What is the average probability of tossing tails again with each of these coins?
In this next part, we begin examining what sorts of outcomes we might
expect from a population that do not all have the same probability of
failure.
In the initial stages of this analysis, we model a team's chances of advancing in the playoffs as being an independent trial - not dependent on the outcome of the playoffs in previous years.
We will also assume, for the moment, that the probability of advancing is constant.
"My shit doesnt't work in the playoffs."
The lament of Billy Beane, oft quoted by his supporters, and held against him by his detractors. The notion that once teams reach the playoffs, the outcome is effectively a crapshoot.
If a team had a 50-50 chance of advancing past the first round of the playoffs, then we ought to expect that a team reaching the playoffs four times will advance twice.
So when a team loses in the fourth round of the playoffs for the fourth consecutive year, is it fair to assume that the team has had some fundamental flaw, that it has failed to advance?
This series will explore this question in some detail.