What is the expectation and sample mean of a random set?

Asked Jul 31 '18 at 01:34

Active Jul 31 '18 at 10:07

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Normally, when I think of a random element $X$ I think of it as being a random variable, or a random vector, or a stochastic process.

Say I have a random element $X$ which instead of being a real value is actually a fixed-cardinality set of real (or perhaps discrete) values, such as:

$$\{1.2, 3.4, 4.5 \}$$ $$\{7.8, 7.7, 6.5\}$$ $$\{10.2,4.5,400.2 \}$$

Is it possible or does it make sense to talk about the expected random set? Does it make sense to talk about higher moments? Practically, If I have observed $n$ i.i.d. sample sets, is it possible to talk about the sample mean of $X$, and other sample moments?

Assume here I don't know how the individual elements interact or what randomness generated them, but I only care about analyzing the set as a whole.

edited Jul 31 '18 at 09:12

asked Jul 31 '18 at 01:34

Ollie

How much measure theory do you know? – Xander Henderson Jul 31 '18 at 01:42
1

Are they really fixed-cardinality sets or are they tuples/sequences? That is, are (1,2,3) and (2,1,3) the same or two different values? Is (2,2,2) allowed? – Jul 31 '18 at 04:34
@Rahul good catch. Order doesn’t matter but repeated values are allowed. So I suppose we are dealing with multisets? – Ollie Jul 31 '18 at 09:09
@XanderHenderson next to nothing – Ollie Jul 31 '18 at 09:10

What is the expectation and sample mean of a random set?

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