2 replaced http://stats.stackexchange.com/ with https://stats.stackexchange.com/ edited Apr 13 '17 at 12:44 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 answered May 10 '13 at 19:53 mrsoltys 14311 silver badge77 bronze badges 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 $$f_Y(y)=\frac{1}{r\pi\sqrt{4-\frac{Y^2}{r^2}}}$$