diff --git a/Exercise sheet 2/exercise_sheet_02.ipynb b/Exercise sheet 2/exercise_sheet_02.ipynb index 1ea15e1..9754ccd 100644 --- a/Exercise sheet 2/exercise_sheet_02.ipynb +++ b/Exercise sheet 2/exercise_sheet_02.ipynb @@ -839,7 +839,7 @@ } }, "source": [ - "The coordinate transformation $T$ is defined as $T(\\Phi, R) = (R\\cos{\\Phi}, R\\sin{\\Phi}) = (X, Y)$. As $T$ is invertible differentiable, we can write the equality between the joint probability density in both coordinate pairs as\n", + "The coordinate transformation $T$ is defined as $T(\\Phi, R) = (R\\cos{\\Phi}, R\\sin{\\Phi}) = (X, Y)$. As $T$ is invertible differentiable, we can write the equality between the joint probability density in both coordinate pairs as follows, using the Jacobian.\n", "$$\n", "f_{X,Y}(x,y) \\Big|\\frac{\\mathrm{d}x}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}y}{\\mathrm{d}r}-\\frac{\\mathrm{d}y}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}x}{\\mathrm{d}r}\\Big|\n", "= f_{X,Y}(T(\\phi,r)) \\Big|\\frac{\\mathrm{d}x}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}y}{\\mathrm{d}r}-\\frac{\\mathrm{d}y}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}x}{\\mathrm{d}r}\\Big|\n",