query: Show that $X_1X_2/\sqrt{X_1^2+X_2^2}$ is normally distributed
positive_passages.docid: 822
positive_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 $N(0,\frac{1}{4})$
negative_passages.docid: 31342
negative_passages.text: How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed?
negative_passages.docid: 103501
negative_passages.text: Question about a normal equation proof
negative_passages.docid: 87411
negative_passages.text: Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$
negative_passages.docid: 91101
negative_passages.text: Proving $\sqrt2$ is irrational
negative_passages.docid: 87055
negative_passages.text: How to prove $\frac{x}{1+x^2}0$?
negative_passages.docid: 122584
negative_passages.text: Converting a Uniform Distribution to a Normal Distribution
negative_passages.docid: 13902
negative_passages.text: Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$.
negative_passages.docid: 54769
negative_passages.text: Distribution of sine of uniform random variable on $[0, 2\pi]$
negative_passages.docid: 48270
negative_passages.text: How to prove that the random variables $X$ and $X^2$ are not independent?
negative_passages.docid: 84203
negative_passages.text: Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ random variables. What is the distribution of $X_1^2 + X_2^2$?
negative_passages.docid: 109295
negative_passages.text: Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$
negative_passages.docid: 127291
negative_passages.text: Show that $\left( \frac{q}{p} \right) \equiv q^{(p-1) / 2} \mod p$, where $\left( \frac{q}{p} \right)$ is the Legendre Symbol
negative_passages.docid: 12184
negative_passages.text: How to prove that Normal Squared Distances follow a Chi-Square distribution?
negative_passages.docid: 116975
negative_passages.text: Given $X$ and $Y$ are independent N(0,1) random variables and $Z = \sqrt{X^2+Y^2}$ from the marginal pdf of $Z$
negative_passages.docid: 66082
negative_passages.text: Using $\sqrt{1-t}\leq 1-\frac t2$ to show that $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}\geq\frac1{2\sqrt n}$
negative_passages.docid: 4265
negative_passages.text: Finding the Moment Generating Function of $X^2$ when $X\sim N(0,1)$