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I dropped two square terms in the derivation for the sum of

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Gaussians; the remaining math was correct.
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Roger Labbe 2017-05-11 13:24:35 -07:00
parent 99193f7e26
commit d2f247da20
1 changed files with 2 additions and 2 deletions
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03-Gaussians.ipynb
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@ -1408,8 +1408,8 @@
"$p(x) = \\int\\limits_{-\\infty}^\\infty f_2(x-x_1)f_1(x_1)\\, dx$\n",
"\n",
"$= \\int\\limits_{-\\infty}^\\infty \n",
"\\frac{1}{\\sqrt{2\\pi}\\sigma_z}\\exp\\left[-\\frac{x - z - \\mu_z}{2\\sigma^2_z}\\right]\n",
"\\frac{1}{\\sqrt{2\\pi}\\sigma_p}\\exp\\left[-\\frac{x - \\mu_p}{2\\sigma^2_p}\\right] \\, dx$\n",
"\\frac{1}{\\sqrt{2\\pi}\\sigma_z}\\exp\\left[-\\frac{(x - z - \\mu_z)^2}{2\\sigma^2_z}\\right]\n",
"\\frac{1}{\\sqrt{2\\pi}\\sigma_p}\\exp\\left[-\\frac{(x - \\mu_p)^2}{2\\sigma^2_p}\\right] \\, dx$\n",
"\n",
"$= \\int\\limits_{-\\infty}^\\infty\n",
"\\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma_p^2 + \\sigma_z^2}} \\exp\\left[ -\\frac{(x - (\\mu_p + \\mu_z)))^2}{2(\\sigma_z^2+\\sigma_p^2)}\\right]\n",