91Ó°ÊÓ

Let \(\mathbf{R}^{2}=\mathbf{R} \times \mathbf{R}\), and let \(\mathcal{B}^{2}\) be the Borel sets of \(\mathbf{R}^{2}\), while \(\mathcal{B}\) denotes the Borel sets of \(\mathbf{R}\). Show that \(\mathcal{B}^{2}=\mathcal{B} \otimes \mathcal{B}\).

Let \(X, Y\) be independent random variables taking values in \(\mathbf{N}\) with $$ P(X=i)=P(Y=i)=\frac{1}{2^{i}} \quad(i=1,2, \ldots) $$ Find the following probabilities: a) \(P(\min (X, Y) \leq i) \quad\left[\right.\) Ans.: \(\left.1-\frac{1}{4^{4}}\right]\) b) \(P(X=Y)\) [Ans.: \(\frac{1}{3}\) ] c) \(P(Y>X) \quad\left[\right.\) Ans.: \(\left.\sum_{i \geq 0}^{3} \frac{1}{2^{2}\left(2^{i}-1\right)}\right]\) d) \(P(X\) divides \(Y)\) [Ans,: \(\frac{1}{3}\) ] e) \(P(X \geq k Y)\) for a given positive integer \(k\) [Ans,: \(\frac{1}{2^{1+k}-1}\) ]