Following on some stats revision, I came up with this interesting scenario:
A random variable X takes nonnegative integer values, and has the property that for 0 < x < y,
P(X > y | X > x) = P(X > y-x)
The form of such a distribution is unique -- what is the distribution function?
Now suppose Y takes nonnegative real values and has the same property. What must be the distribution of Y?
Now suppose the density function of Y takes the same value at y=0 as the mass function of X does at x=0.
Now let Z = floor(Y). Z is a random variable with the same range as X. What is the distribution function of Z? Is Z 'forgetful' in the same sense that X and Y are?
Finally, consider W = X+Z. Is W forgetful in the above sense? Informationally speaking, what does this mean?
Note that some of the detail is extraneous, but this way you can also see how this forgetfulness property differs in discrete and continuous situations.