2 replaced http://stats.stackexchange.com/ with https://stats.stackexchange.com/
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Thanks for the help. @whuber pointed me to a post discussing changing variablesdiscussing changing variables, which lead me to read a bit on wikipedia.

a change in variables is defined as: $f_Y(y) = \left| \frac{d}{dy} (g^{-1}(y)) \right| \cdot f_X(g^{-1}(y))$

For the example above:

$Y=2 r sin(Θ/2)$

$Θ=2 arcsin{\frac{Y}{2r}}$

$\frac{dΘ}{dY}=\frac{2}{r\sqrt{4-\frac{Y^2}{r^2}}}$, $f(θ)=1/2π$

so $f_Y(y)=\frac{1}{r\pi\sqrt{4-\frac{Y^2}{r^2}}}$

Thanks for the help. @whuber pointed me to a post discussing changing variables, which lead me to read a bit on wikipedia.

a change in variables is defined as: $f_Y(y) = \left| \frac{d}{dy} (g^{-1}(y)) \right| \cdot f_X(g^{-1}(y))$

For the example above:

$Y=2 r sin(Θ/2)$

$Θ=2 arcsin{\frac{Y}{2r}}$

$\frac{dΘ}{dY}=\frac{2}{r\sqrt{4-\frac{Y^2}{r^2}}}$, $f(θ)=1/2π$

so $f_Y(y)=\frac{1}{r\pi\sqrt{4-\frac{Y^2}{r^2}}}$

Thanks for the help. @whuber pointed me to a post discussing changing variables, which lead me to read a bit on wikipedia.

a change in variables is defined as: $f_Y(y) = \left| \frac{d}{dy} (g^{-1}(y)) \right| \cdot f_X(g^{-1}(y))$

For the example above:

$Y=2 r sin(Θ/2)$

$Θ=2 arcsin{\frac{Y}{2r}}$

$\frac{dΘ}{dY}=\frac{2}{r\sqrt{4-\frac{Y^2}{r^2}}}$, $f(θ)=1/2π$

so $f_Y(y)=\frac{1}{r\pi\sqrt{4-\frac{Y^2}{r^2}}}$

1
source | link

Thanks for the help. @whuber pointed me to a post discussing changing variables, which lead me to read a bit on wikipedia.

a change in variables is defined as: $f_Y(y) = \left| \frac{d}{dy} (g^{-1}(y)) \right| \cdot f_X(g^{-1}(y))$

For the example above:

$Y=2 r sin(Θ/2)$

$Θ=2 arcsin{\frac{Y}{2r}}$

$\frac{dΘ}{dY}=\frac{2}{r\sqrt{4-\frac{Y^2}{r^2}}}$, $f(θ)=1/2π$

so $f_Y(y)=\frac{1}{r\pi\sqrt{4-\frac{Y^2}{r^2}}}$