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.
There are a number of different possible distributions to consider. In each case, the analysis is much the same. First, the distribution is defined on some interval [M-a, M+a].
The definition will feature an arbitrary multiplicative constant in front. So the second step is to calculate the value of this constant - which is determined by recognizing that the integral satisfies the property
∫ P(x) dx = 1
In the distributions we'll be looking at, these constants will be trivial to calculate.
Having done this, the exercise is parallel to the discrete case. We first calculate ∫ x^4 P(x)dx, determining how many times we should expect that four flips all land tails. We then calculate ∫ x^5 P(x)dx, which combined with the previous result determines the expected probability of landing tails of those coins that did so in all four trials.
Because we are using integrals, rather than sums, the cancellation of terms differs slightly. Taking the integral promotes the exponent - we'll be looking at (M+a)^5, where we had previously seen (M+a)^4 terms; also, we'll be subtracting, rather than adding, the (M-a)^5 term. As in the previous case, this eliminates the odd power terms; the familiar factor of 1/2 keeps the even terms simple.
There will typically be two additional cancellations. The process of taking the integral will produce a number in the denominator which cancels the coefficient of the leading power of the expansion. The normalization constant will cancel part of the factor of the leading term, so the equations will reduce to those already seen in the limiting case.
May 11, 2005 8:08 PM
| TrackBack