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08: Solve a and b and add a TODO

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Kees van Kempen
2022-11-15 09:01:41 +01:00
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Exercise sheet 8/exercise_sheet_08.ipynb
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@ -421,7 +421,38 @@
}
},
"source": [
"YOUR ANSWER HERE"
"**To proof**\n",
"\n",
"$\\frac{1}{V}\\mathbb{E}[ \\rho_T(r) ] = \\mathbb{P}(d_T(X,Y) = r)$\n",
"\n",
"**Proof**\n",
"\n",
"$$\n",
"\\frac{1}{V} \\mathbb{E}\\left[ \\rho_T(r)\\right]\n",
" = \\frac{1}{V} \\mathbb{E} \\left[\\frac{1}{V} \\sum_{x=0}^{V-1}\\sum_{y=0}^{V-1} \\mathbf{1}_{\\{d_T(x,y)=r\\}} \\right]\n",
" = \\frac{1}{V^2} \\mathbb{E} \\left[ \\sum_{x=0}^{V-1}\\sum_{y=0}^{V-1} \\mathbf{1}_{\\{d_T(x,y)=r\\}} \\right]\n",
" = \\frac{1}{V^2} \\sum_{x=0}^{V-1}\\sum_{y=0}^{V-1} \\mathbb{E} \\left[ \\mathbf{1}_{\\{d_T(x,y)=r\\}} \\right]\n",
"$$\n",
"\n",
"The order of summation is changed, as the sum of expectation values is equal to the expectation value of the sum.\n",
"The latter expectation value of the indicator function is exactly equal to the chance $\\mathbb{P}(d_T(x,y)=r)$ for given $x, y$.\n",
"For the uniformly distributed $X, Y$, we find $\\mathbb{P}(X = x) = \\frac{1}{V} = \\mathbb{P}(Y = y)$.\n",
"This allows us to write the right hand side as follows.\n",
"\n",
"$$\n",
"\\frac{1}{V} \\mathbb{E}\\left[ \\rho_T(r)\\right]\n",
" = \\frac{1}{V^2} \\sum_{x=0}^{V-1}\\sum_{y=0}^{V-1} \\mathbb{P}(d_T(x,y)=r)\n",
" = \\sum_{x=0}^{V-1}\\sum_{y=0}^{V-1} \\mathbb{P}(X = x) \\mathbb{P}(Y = y) \\mathbb{P}(d_T(x,y)=r)\n",
" = \\mathbb{P}(d_T(X,Y)=r),\n",
"$$\n",
"\n",
"which is what we sought.\n",
"\n",
"Using this result, it is just a matter of writing out the definition of an expectation value to get to the result.\n",
"\n",
"$$\n",
"\\mathbb{E}[d_T(X,Y)] = \\sum_{r=0}^\\infty r\\, \\mathbb{P}(d_T(X,Y) = r) = \\frac{1}{V}\\sum_{r=0}^\\infty r\\, \\mathbb{E}[ \\rho_T(r) ].\n",
"$$"
]
},
{
@ -475,7 +506,36 @@
}
},
"source": [
"YOUR ANSWER HERE"
"**To proof**\n",
"\n",
"$$\n",
"\\mathbb{E}[d_T(X,Y)] \\approx c\\,V^{1/d_H}, \\qquad c = \\int_0^\\infty \\mathrm{d}x\\,x\\,f(x)\n",
"$$\n",
"\n",
"**Proof**\n",
"\n",
"$$\n",
"\\mathbb{E} \\left[ d_T(X,Y) \\right]\n",
" = \\frac{1}{V} \\sum_{r=0}^\\infty r\\, \\mathbb{E} \\left[ \\rho_T(r) \\right]\n",
" = \\frac{1}{V} \\sum_{r=0}^\\infty rV^{1-1/d_H}f(rV^{-1/d_H})\n",
" = \\frac{1}{V} \\sum_{r=0}^\\infty xV^{1/d_H} \\cdot V^{1-1/d_H}f(x)\n",
" = \\sum_{r=0}^\\infty xf(x),\n",
"$$\n",
"where the first equality sign is due to (2), the second due to the given assumption, the third using $x = rV^{-1/d_H}$.\n",
"\n",
"Now we approximate the summation by an integral.\n",
"\n",
"$$\n",
"\\sum_{r=0}^\\infty xf(x)\n",
" \\approx \\int_{r=0}^\\infty xf(x)dr\n",
" = V^{1/d_H} \\int_{x=0}^\\infty xf(x)dx\n",
" = cV^{1/d_H},\n",
"$$\n",
"using $\\frac{dr}{dx} = V^{1/d_H}$ for substitution.\n",
"This yields the desired approximation\n",
"$$\n",
" \\mathbb{E} \\left[ d_T(X,Y) \\right] \\approx cV^{1/d_H}.\n",
"$$"
]
},
{
@ -602,6 +662,7 @@
" \n",
" for idx_model, model in enumerate(models):\n",
" # Calculate mean and standard deviation of the expectations.\n",
" # TODO: Look at whether I store the right data and do the right calculations.\n",
" mu = np.mean(expectations[model], 1)\n",
" sigma = np.std(expectations[model], 1)\n",
"\n",