A puzzle for jay_shark et al.

[Siegmund]

One of my personal favourites - actually, one of very few bits of mathematical trivia I discovered for myself and never subsequently ran across in a book anywhere.

Let X be a point selected at random from the standard middle-thirds-deleted Cantor Set on [0,1]. What is Var(X)?

[jason1990]

<font color="white">I would construct a random element from the Cantor set as follows. Let X_n be an iid sequence with P(X_n = 0) = P(X_n = 1) = 0.5, and define the random element as

X = \sum_{n=1}^\infty 2X_n/3^n.

We then have

Var(2X_n/3^n) = (4/9^n)Var(X_n) = 1/9^n.

Since the summands are independent, the variances add, giving

Var(X) = \sum_{n=1}^\infty 1/9^n = 1/8.

(The Dominated Convergence Theorem justifies adding the variances all the way to infinity.)</font>

[bxb]

I got 1/8.

[Siegmund]

You're right. Both of you, that is. Too easy