Forgetfulness

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.

Originally posted by royalchicken
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 p ...[text shortened]... ou can also see how this forgetfulness property differs in discrete and continuous situations.

I suppose < means 'less or equal', and similarly > means 'greater or equal' in this context?

Originally posted by TheMaster37
I suppose < means 'less or equal', and similarly > means 'greater or equal' in this context?

No, the inequalities are strict, although it doesn't really matter in the continuous case.

They are strict so that expressions like P(X > x) can be written in terms of distribution functions: P(X > x) = 1 - F(x) where F is the cdf of X.

Thanks for looking at this.

Originally posted by royalchicken
No, the inequalities are strict, although it doesn't really matter in the continuous case.

They are strict so that expressions like P(X > x) can be written in terms of distribution functions: P(X > x) = 1 - F(x) where F is the cdf of X.

Thanks for looking at this.

Ah, I was thinking too much of a recursive formula :p Silly me.

I'll try to look at this, but these kind of things aren't my thing :p

Originally posted by TheMaster37
Ah, I was thinking too much of a recursive formula :p Silly me.

I'll try to look at this, but these kind of things aren't my thing :p

Sure, I'm glad to see any sort of interest at all being taken 🙂.