Re: Improper distribution



On 26 Jan, 22:59, "Michael" <mchlg...@xxxxxxx> wrote:
Assume I wish to write down the distribution (probability density)
of times I take for a given journey that I repeatedly make.
Given that I may have an accident during the journey, and die, and
so never complete the journey, there is a finite probability that I
will
take an infinite amount of time to complete the journey.

My question is: is there a standard way of setting up the distribution
in this case, for a continuous random variable, where the infinite
value has a non-negligible probability.You have a mixture between a continuous and a discrete distribution.
I've seen that handled with Dirac delta functions; yours would be at
infinity.

So your probability density function would be something like this:
p(x) = f(x) + p_inf * delta(x - infinity)
where f(x) is the regular old continuous part and p_inf is the
point-mass probability at infinity.

You can confirm that the integral from -infinity to infinity is 1 and
all that.

Mixing continous and discrete is not pretty, but it works, provided
you're careful with the math.

The other way to do it would be consider P[finite] and P[infinite] (a
discrete distribution of 2 numbers) and p(x | finite) = C * f(x), p (x
| infinite) = 0 (x < infinity), and handle this all as a combination of
two random variables, one conditioned on the other. (Here C is the
appropriate normalization constant.)

Michael

Michael,

Intuitively, I would expect something like a delta function at
infinity,
But does that have any meaning? When does it contribute to an
integral i.e. an expectation, given the way an integral is defined,
as a limit towards the upper end point.

Inf

.

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