From 732490e34b1b034341afb62bafcfbb8101b9629f Mon Sep 17 00:00:00 2001
From: Kees van Kempen <k.vankempen@student.science.ru.nl>
Date: Mon, 19 Sep 2022 21:51:04 +0200
Subject: [PATCH] 02: Perform the coordinate transformation

---
 Exercise sheet 2/exercise_sheet_02.ipynb | 13 ++++++++++++-
 1 file changed, 12 insertions(+), 1 deletion(-)

diff --git a/Exercise sheet 2/exercise_sheet_02.ipynb b/Exercise sheet 2/exercise_sheet_02.ipynb
index c924c1c..201979f 100644
--- a/Exercise sheet 2/exercise_sheet_02.ipynb	
+++ b/Exercise sheet 2/exercise_sheet_02.ipynb	
@@ -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 $\\mathcal{N}(0,1)$"
    ]
   },
   {