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    "---\n",
    "title: Biostat 216 Homework 1\n",
    "subtitle: 'Due Oct 4 @ 11:59pm'\n",
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    "Submit a PDF (scanned/photographed from handwritten solutions, or converted from RMarkdown or Jupyter Notebook or Quarto) to Gradescope in BruinLearn. \n",
    "\n",
    "## Q1. Average and norm\n",
    "\n",
    "Use the Cauchy-Schwarz inequality to prove that\n",
    "$$\n",
    "- \\frac{1}{\\sqrt{n}} \\|\\mathbf{x}\\| \\le \\frac{1}{n} \\sum_{i=1}^n x_i \\le \\frac{1}{\\sqrt{n}} \\|\\mathbf{x}\\|\n",
    "$$\n",
    "for any $\\mathbf{x} \\in \\mathbb{R}^n$. In other words, $-\\operatorname{rms}(\\mathbf{x}) \\le \\operatorname{avg}(\\mathbf{x}) \\le \\operatorname{rms}(\\mathbf{x})$.\n",
    "\n",
    "What are the conditions on $\\mathbf{x}$ to have equality in the upper bound? When do we have equality in the lower bound?\n",
    "\n",
    "## Q2. AM-HM inequality\n",
    "\n",
    "Use the Cauchy-Schwartz inequality to prove that\n",
    "$$\n",
    "\\frac{1}{n} \\sum_{i=1}^n x_i \\ge \\left( \\frac{1}{n} \\sum_{i=1}^n \\frac{1}{x_i} \\right)^{-1}\n",
    "$$\n",
    "for any $\\mathbf{x} \\in \\mathbb{R}^n$ with positive entries $x_i$. \n",
    "\n",
    "The left hand side is called the arithmetic mean (AM) and the right hand side is called the harmonic mean (HM). You may wonder what can be a practical use of such a simple inequality. See this [paper](http://hua-zhou.github.io/media/pdf/LangeZhou14GP.pdf), which uses the AM-HM inequality to derive a class of optimization algorithms for geometric and signomial programming.\n",
    "\n",
    "## Q3. Bias-variance tradeoff\n",
    "\n",
    "Prove the formula \n",
    "$$\n",
    "\\operatorname{avg}(\\mathbf{x})^2 + \\operatorname{std}(\\mathbf{x})^2 = \\operatorname{rms}(\\mathbf{x})^2\n",
    "$$\n",
    "using the vector notation and do BV 3.15.\n",
    "\n",
    "## BV exercises\n",
    "\n",
    "1.7, 1.9, 1.13, 1.16, 1.20, 3.4, 3.5, 3.12."
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