91Ó°ÊÓ

Consider a theatre with \(n\) seats that is fully booked for this evening. Each of the \(n\) people entering the theatre (one by one) has a seat reservation. However, the first person is absent-minded and takes a seat at random. Any subsequent person takes his or her reserved seat if it is free and otherwise picks a free seat at random. (i) What is the probability that the last person gets his or her reserved seat? (ii) What is the probability that the \(k\) th person gets his or her reserved seat?

Let \(p, q \in(1, \infty)\) with \(\frac{1}{p}+\frac{1}{q}=1\) and let \(X \in \mathcal{L}^{p}(\mathbf{P})\) and \(Y \in \mathcal{L}^{q}(\mu)\). Let \(\mathcal{F} \subset \mathcal{A}\) be a \(\sigma\)-algebra. Use the preceding theorem to show the conditional version of Hölder's inequality: $$ \mathbf{E}[|X Y| \mid \mathcal{F}] \leq \mathbf{E}\left[|X|^{p} \mid \mathcal{F}\right]^{1 / p} \mathbf{E}\left[|Y|^{q} \mid \mathcal{F}\right]^{1 / q} \quad \text { almost surely. } $$

Show the conditional Markov inequality: For monotone increasing \(f:[0, \infty) \rightarrow[0, \infty)\) and \(\varepsilon>0\) with \(f(\varepsilon)>0\) $$ \mathbf{P}[|X| \geq \varepsilon \mid \mathcal{F}] \leq \frac{\mathbf{E}[f(|X|) \mid \mathcal{F}]}{f(\varepsilon)}. $$

Assume the random variable \((X, Y)\) is uniformly distributed on the disc \(B:=\left\\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leq 1\right\\}\) and on \([-1,1]^{2}\), respectively. (i) In both cases, determine the conditional distribution of \(Y\) given \(X=x\). (ii) Let \(R:=\sqrt{X^{2}+Y^{2}}\) and \(\Theta=\arctan (Y / X) .\) In both cases, determine the conditional distribution of \(\Theta\) given \(R=r\).

Show the conditional Cauchy-Schwarz inequality: For square integrable random variables \(X, Y\), $$ \mathbf{E}[X Y \mid \mathcal{F}]^{2} \leq \mathbf{E}\left[X^{2} \mid \mathcal{F}\right] \mathbf{E}\left[Y^{2} \mid \mathcal{F}\right]. $$