Let \(X_1, X_2, X_3, \ldots\) be independent random variables that are all uniform on \([0,1]\text{.}\) Prove that the following sequences of random variables converge in probability and find the constant they converge to. (You may use without proof that \(\E(X_i) = \frac{1}{2}\) and \(\var(X_i) = \frac{1}{12}\text{.}\))
For part (c): The random variables \(X_iX_{i+1}\) and \(X_{i+1}X_{i+2}\) are not independent, so their covariance is not necessarily 0. You should be careful when computing \(\var(Y_n)\text{.}\)
You arrange \(n \gt 2\) cubical boxes of side length 1 numbered \(1, 2, \ldots, n\) in a circle in clockwise order. Independently generate a random number \(X_i \sim \uniform([0,1])\) for each \(1 \leq i \leq n\text{,}\) and then fill box \(i\) up to height \(X_i\) with sand. We say that box \(i\) is a gap box if it’s filled at least to height 3/4, and the next box in clockwise order is filled to height at most 1/4. Let \(N_n\) be the number of gap boxes.
If boxes \(i\) and \(j\) are separated by at least one box, then the events “box \(i\) is a gap box” and “box \(j\) is a gap box” are independent. Use this to compute \(\var(N_n)\text{.}\)
Suppose that you have a total of \(2n\) cards consisting of \(n\) pairs of cards where each pair is a different color. Select \(n\) cards without replacement and let \(N_n\) denote the number of colors that are not present in your selection.
