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02: Perform the coordinate transformation

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Kees van Kempen
2022-09-19 21:51:04 +02:00
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Exercise sheet 2/exercise_sheet_02.ipynb
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@ -839,7 +839,18 @@
}
},
"source": [
"YOUR ANSWER HERE"
"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",
"$$\n",
"f_{X}(x)f_Y(y) \\Big|\\frac{\\mathrm{d}x}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}y}{\\mathrm{d}r}-\\frac{\\mathrm{d}y}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}x}{\\mathrm{d}r}\\Big|\n",
"= f_{X,Y}(x,y) \\Big|\\frac{\\mathrm{d}x}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}y}{\\mathrm{d}r}-\\frac{\\mathrm{d}y}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}x}{\\mathrm{d}r}\\Big|\n",
"= f_{X,Y}(T(\\phi,r)) \\Big|\\frac{\\mathrm{d}x}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}y}{\\mathrm{d}r}-\\frac{\\mathrm{d}y}{\\mathrm{d}\\phi}\\frac{\\mathrm{d}x}{\\mathrm{d}r}\\Big|\n",
"= f_{X,Y}(T(\\phi,r)) \\Big|-r\\sin{\\phi}\\sin{\\phi}-r\\cos{\\phi}\\cos{\\phi}\\Big|\n",
"= f_{X,Y}(T(\\phi,r)) r\n",
"= f_{\\Phi,R}(\\phi,r)\n",
"= \\frac{1}{2\\pi}\\, r\\,e^{-r^2/2}\n",
"\\\\ \\implies f_{X}(x)f_Y(y) = \\frac{1}{2\\pi}\\,e^{-r^2/2} = \\frac{1}{\\sqrt{2\\pi}}e^{-x^2/2}\\frac{1}{\\sqrt{2\\pi}}e^{-y^2/2}\n",
"$$\n",
"using that $r^2 = x^2 + y^2$ in the last step. We conclude that $X$ and $Y$ are independent, and that they are both distributed as a standard normal distribution $\\mathcal{N}(0,1)$"
]
},
{