Suppose we have a stick of length L. We break it once at some point X ~ Unif(0;L). Then we break it again at some point Y ~ Unif(0;X). Use the law of iterated expectation to calculate E[Y].
I'm not sure how to do this problem. I know that E[Y] = E[E[Y|L]] but I don't know what to do with it. Please help! Thanks!
Hi, this is an interesting and tricky little problem.
The law of iterated expectations, also called the law of total expectation (which I like better because it hints at its connections to the law of total probability), is an amazingly powerful tool in probability/statistics.
If I understand your question correctly, we're looking for the expected value (i.e. expected length) of the stick after we've broken it twice.
So, if we call X the length after the first break and Y the length after the second break, then we can say that:
\[ E(Y) = E \big( E(Y\,|\,X)\big) \]
Because we're dealing with a uniform distribution, when we choose the first point where we'll break the stick the expected value of the length of the resulting stick is just half the length of the original stick, i.e. half of L (because every point on the stick is equally likely to be chosen as the breaking point). So we can say that:
\[E(X) = \tfrac{1}{2}L \]
Likewise, when we choose the next point where we break the stick (which now has length X), every point on this stick is equally likely to be selected. So the expected value of Y given a stick of length X is just half the length of the stick before breaking it again, i.e half of X:
\[ E(Y \,|\,X) = \tfrac{1}{2}X \]
Can you see where to take it from here?
Hope that helps!
