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Corrected the claimed formula
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John Jiang
John Jiang
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ConsiderLet $X_i, Y_i, Z_i$ be three independenceiid family of standard normal random variables $X,Y,Z$, where $X,Z$ are each distributed as. Then the first coordinate of a unit circle in $\mathbb{R}^2$ and $Y$following random variable is distributed the same as the first coordinate of a unit $S^2$ inuniform $\mathbb{R}^3$$2$-sphere:

$$ \frac{X_1}{\sqrt{X_1^2+X_2^2}} \frac{Y_1}{\sqrt{Y_1^2+Y_2^2}} + \frac{X_2}{\sqrt{X_1^2+X_2^2}} \frac{Y_2}{\sqrt{Y_1^2+Y_2^2}} \frac{Z_1}{\sqrt{Z_1^2+Z_2^2+Z_3^2}}$$. Then it is true that

Of course, $XZY + XZ$ is distributed as the first coordinate in$\frac{X_1}{\sqrt{X_1^2+X_2^2}}$ squared has $S^2$ again!$\beta(1/2,1/2)$ distribution.

I know this is true by looking at the matrix product $\gamma_{1,2}(\theta_1) \gamma_{2,3}(\theta_2) \gamma_{1,3}(\theta_3)$ where $\gamma_{i,j}(\theta)$ is the rotation matrix by $\theta$ in the $i \wedge j$ plane in $\mathbb{R}^3$. But is there a probabilistic way to prove it using beta-gamma algebra or otherwise?

Consider three independence random variables $X,Y,Z$, where $X,Z$ are each distributed as the first coordinate of a unit circle in $\mathbb{R}^2$ and $Y$ is distributed as the first coordinate of a unit $S^2$ in $\mathbb{R}^3$. Then it is true that $XZY + XZ$ is distributed as the first coordinate in $S^2$ again!

I know this is true by looking at the matrix product $\gamma_{1,2}(\theta_1) \gamma_{2,3}(\theta_2) \gamma_{1,3}(\theta_3)$ where $\gamma_{i,j}(\theta)$ is the rotation matrix by $\theta$ in the $i \wedge j$ plane in $\mathbb{R}^3$. But is there a probabilistic way to prove it using beta-gamma algebra or otherwise?

Let $X_i, Y_i, Z_i$ be three iid family of standard normal random variables. Then the following random variable is distributed the same as the first coordinate of a uniform $2$-sphere:

$$ \frac{X_1}{\sqrt{X_1^2+X_2^2}} \frac{Y_1}{\sqrt{Y_1^2+Y_2^2}} + \frac{X_2}{\sqrt{X_1^2+X_2^2}} \frac{Y_2}{\sqrt{Y_1^2+Y_2^2}} \frac{Z_1}{\sqrt{Z_1^2+Z_2^2+Z_3^2}}$$.

Of course, $\frac{X_1}{\sqrt{X_1^2+X_2^2}}$ squared has $\beta(1/2,1/2)$ distribution.

I know this is true by looking at the matrix product $\gamma_{1,2}(\theta_1) \gamma_{2,3}(\theta_2) \gamma_{1,3}(\theta_3)$ where $\gamma_{i,j}(\theta)$ is the rotation matrix by $\theta$ in the $i \wedge j$ plane in $\mathbb{R}^3$. But is there a probabilistic way to prove it using beta-gamma algebra or otherwise?

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John Jiang
John Jiang
  • 4.5k
  • 25
  • 47

some weird relations among beta random variables

Consider three independence random variables $X,Y,Z$, where $X,Z$ are each distributed as the first coordinate of a unit circle in $\mathbb{R}^2$ and $Y$ is distributed as the first coordinate of a unit $S^2$ in $\mathbb{R}^3$. Then it is true that $XZY + XZ$ is distributed as the first coordinate in $S^2$ again!

I know this is true by looking at the matrix product $\gamma_{1,2}(\theta_1) \gamma_{2,3}(\theta_2) \gamma_{1,3}(\theta_3)$ where $\gamma_{i,j}(\theta)$ is the rotation matrix by $\theta$ in the $i \wedge j$ plane in $\mathbb{R}^3$. But is there a probabilistic way to prove it using beta-gamma algebra or otherwise?