From Osher Doctorow
The Dirac Mass basically comes from the Dirac Delta-Function which has
the values:
1) delta(x) = infinity if x = 0 (i.e., a specified point)
0 otherwise
In Probable Causation/Influence (PI), expectation or expected value
("(population) mean") E(X-->Y) for X, Y continuous random variables is
calculated in the "spirit" of conditional expectation E(Y|X=x) where
the latter is:
2) I(y fY|X(y|x))dy, I...dy being integral, fY|X being conditional
pdf
However, instead of using the conditional pdf fY|X=x, we use in PI
expectation the quantity fX-->Y which is:
3) fX-->Y(x,y) = 1 + f(x,y) - fX(x)
recalling that fY|X=x is f(x,y) divided by fX(x).
So PI expectation is:
4) E(X-->Y) = I(y fX-->Y(x,y))dy
which from (3) turns out to have 3 different values or ranges of
values depending on whether the random variables X and Y are
continuous nonzero only on a finite interval, are positive on the
whole positive real line (asymmetric), or are symmetric on the whole
real line respectively:
5) E(X-->Y) = (1 - fX(x))(b^2 - a^2)/2 + fX(x)E(Y|X=x) where X, Y are
nonzero only on finite interval [a, b]
E(X-->Y) = infinity if X, Y are asymmetric (have asymmetric pdfs)
nonzero on the positive real axis.
E(X-->Y) is undefined if X, Y are symmetric (have symmetric pdfs)
on the whole real line.
The similarity to the infinity case between the Dirac Delta Function
and E(X-->Y) for the asymmetric real line case is noteworthy,
especially because some of the most important probability
distributions are asymmetric including:
6) the gamma distribution (with subtypes chi-square and exponential
distributions among many others)
7) the F distribution used in scientific statistical experiments and
linear/multilinear regression analysis.
Thus, in a sense, E(X-->Y) singles out positive half real line
asymmetric probability distributions as key instead of "point masses"
like x = 0.
The first case in (5) is somewhat analogous to the non-0 x (but 0
value of the Dirac Function) case of the Dirac Delta Function since it
is the only other defined case of (5) and interpolates between (b^2 -
a^2)/2 and E(Y|X=x) noting that fX(x) and 1 - fX(x) are always between
0 and 1 except for extremely high-peaked distributions such as very
narrow Gaussian/normal distributions. However, it does single out the
very important Uniform and Beta and Power probability distributions,
while the Dirac Delta Function ignores everything outside the single
point mass so to speak.
The last case in (5), the undefined symmetric case, has no analog in
the Dirac Delta Function.
Osher Doctorow
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