The grass in my garden grows by 1cm every week. Every Sunday morning I cut the grass by selecting half of the blades independently at random and cutting them to zero length. Assuming I have been doing this for a long time what is the average length of the blades in my garden?
Since the problem is recursive one way to solve it I though was to say. If $X_i$ representes the length of the blades in the garden after the $i$-th Saturday, then we have:
$$X_{i+1} = 0.5 \times (X_i +1) + 0.5 \times (X \times 0) = 0.5 \times (X_i +1)$$
So assuming the expectation converges after a long time by solving the equation I found $\mathbb{E}[X] = 1$.
Though I am not convinced by what I am doing... Like what does $X$ actually represents and on what is it defined?
A proper way of defining it is assuming the garden is a square and then we defined: $X:[0,1]^2 \mapsto \mathbb{R}_{+}$. So for each blade in the garden $X$ gives its length.
But then we need to define a random variable that takes randomly half of the points in: $[0, 1]^2$. Hence I really don't know how to solve this problem properly because I feel like in my argument above something is completely missing.
Is there a way of solving this problem properly? And I am right by saying that the first argument above is not rigourously correct?