query: Prove independence of $Y_0$ and $\sigma$-algebra $\mathscr{R}$.
positive_passages.docid: 143732
positive_passages.text: Prove random variable $Y_0$ and $\sigma$-algebra $\mathscr{R}$ are independent.
negative_passages.docid: 63860
negative_passages.text: Measure of Dependence Between Two $\sigma$-algebras
negative_passages.docid: 86717
negative_passages.text: Completeness of $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle$
negative_passages.docid: 142594
negative_passages.text: Verifying the interpretation of stopping times and stopping time $\sigma$-algebras
negative_passages.docid: 121853
negative_passages.text: $X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$
negative_passages.docid: 152495
negative_passages.text: Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent
negative_passages.docid: 40609
negative_passages.text: Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
negative_passages.docid: 92309
negative_passages.text: The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$
negative_passages.docid: 64118
negative_passages.text: Number of elements in a finite $\sigma$-algebra
negative_passages.docid: 51134
negative_passages.text: If $E \in \sigma(\mathcal{C})$ then there exists a countable subset $\mathcal{C}_0 \subseteq \mathcal{C}$ with $E \in \sigma(\mathcal{C}_0)$
negative_passages.docid: 135641
negative_passages.text: Algebraic independence of exponentials
negative_passages.docid: 147783
negative_passages.text: Showing a sigma algebra is the union of certain collection of sigma algebras
negative_passages.docid: 19973
negative_passages.text: If $X$ and $Y$ are independent $N(0,\sigma^2)$, then $X^2+Y^2$ and $X/Y$ are independent?
negative_passages.docid: 49806
negative_passages.text: Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$
negative_passages.docid: 84437
negative_passages.text: Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$
negative_passages.docid: 142325
negative_passages.text: Two independent geometric random variables - proof of sum
negative_passages.docid: 74162
negative_passages.text: Show intersection of two algebras are not a $\sigma$-algebra