query: If I have that $X,Y,Z$ are jointly independent random variables, how can I show that $X$ is independent of $YZ$ (the product)?
positive_passages.docid: 147866
positive_passages.text: If X,Y and Z are independent, are X and YZ independent?
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: 30500
negative_passages.text: Independence of ratios of independent variates
negative_passages.docid: 65784
negative_passages.text: If $X,Y$ are independent $\chi ^2$ with $m$ and $n$ degrees of freedom, then $\frac{X}{X+Y} \sim\beta(m/2, n/2)$
negative_passages.docid: 14664
negative_passages.text: Sum of indepedent random variables and a constant
negative_passages.docid: 103780
negative_passages.text: Independence of disjoint events
negative_passages.docid: 86729
negative_passages.text: Are squares of independent random variables independent?
negative_passages.docid: 135085
negative_passages.text: The mutual density of $X,Y$ in $\{|t|+|s|<1\}$ is constant, are $X,Y$ independent?
negative_passages.docid: 25654
negative_passages.text: E[X| X>Y] for independent X, Y ~ N(0,1)
negative_passages.docid: 48270
negative_passages.text: How to prove that the random variables $X$ and $X^2$ are not independent?
negative_passages.docid: 29339
negative_passages.text: Are functions of independent variables also independent?
negative_passages.docid: 39378
negative_passages.text: Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.
negative_passages.docid: 43181
negative_passages.text: What is the joint distribution of $Z=\min(X,Y)$ and $I_{Z=X}$?
negative_passages.docid: 79457
negative_passages.text: Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :
negative_passages.docid: 85790
negative_passages.text: Product of two random variables that has exponential distribution
negative_passages.docid: 117907
negative_passages.text: A question regarding independence of $\min{\{X,Y\}}$ and $X-Y$ when $X,Y$ follows iid geometric distribution
negative_passages.docid: 822
negative_passages.text: Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$