Let \(X=(X_i)_{i=1}^{\infty}\) be an iid sequence of random variables. Let \(N\) be a random nonnegative integer that is independent of \(X\). Suppose \(X_1\) and \(N\) have finite expectations. Prove that \[ \mathbb{E} \big( \sum_{i=1} ^N X_i \big) = \mathbb{E}N \mathbb{E} X_1\]
How would you do Exercise 4 analytically?
What would you do if there is more than two tiles, and the tiles did not occur with equal probability?
Use simulations to test your formula.
Let \(\Pi\) be a Poisson point process on \([0, \infty)\). Pick a (large) number, say \(x=\sqrt{2} + 100\). Find the smallest interval \((A,B)\) such that \(x \in (A,B)\), and \(A\) and \(B\) are points of \(\Pi\), thus \(B\) is the next arrival after \(A\).
Find the distribution of \(S=B - A\).
Is \(S\) exponentially distributed? Explain.
Do simulations to confirm your findings.