Chapter 05 - Inequalities.ipynb

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   "source": [
    "## 5. Inequalities"
   ]
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   "source": [
    "### 5.1 Markov and Chebyshev Inequalities"
   ]
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   "source": [
    "**Theorem 5.1 (Markov's Inequality)**.  Let $X$ be a non-negative random variable and suppose that $\\mathbb{E}(X)$ exists.  For any $t > 0$,\n",
    "\n",
    "$$ \\mathbb{P}(X > t) \\leq \\frac{\\mathbb{E}(X)}{t} $$"
   ]
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    "**Proof**.\n",
    "\n",
    "$$ \n",
    "\\mathbb{E}(X)\n",
    "=\\int_0^\\infty xf(x) dx\n",
    "=\\int_0^t xf(x) dx + \\int_t^\\infty xf(x) dx\n",
    "\\geq \\int_t^\\infty xf(x) dx\n",
    "\\geq t \\int_t^\\infty f(x) dx\n",
    "= t \\mathbb{P}(X > t)\n",
    "$$"
   ]
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   "source": [
    "**Theorem 5.2 (Chebyshev's Inequality)**.  Let $\\mu = \\mathbb{E}(X)$ and $\\sigma^2 = \\mathbb{V}(X)$.  Then,\n",
    "\n",
    "$$ \\mathbb{P}(|X - \\mu| \\geq t) \\leq \\frac{\\sigma^2}{t^2} \n",
    "\\quad \\text{and} \\quad\n",
    "\\mathbb{P}(|Z| \\geq k) \\leq \\frac{1}{k^2} $$\n",
    "\n",
    "where $Z = (X - \\mu) / \\sigma$.  In particular, $\\mathbb{P}(|Z| > 2) \\leq 1/4$ and $\\mathbb{P}(|Z| > 3) \\leq 1/9$."
   ]
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   "source": [
    "**Proof**.  We use Markov's inequality to conclude that\n",
    "\n",
    "$$ \\mathbb{P}(|X - \\mu| \\geq t) = \\mathbb{P}(|X - \\mu|^2 \\geq t^2) \\leq \\frac{\\mathbb{E}(X - \\mu)^2}{t^2} = \\frac{\\sigma^2}{t^2} $$\n",
    "\n",
    "The second part follows by setting $t = k \\sigma$."
   ]
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   "source": [
    "### 5.2 Hoeffding's Inequality"
   ]
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   "source": [
    "Hoeffding's inequality is similar in spirit to Markov's inequality but it is a sharper inequality.  We present the result here in two parts.  The proofs are in the technical appendix."
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   "source": [
    "**Theorem 5.4**.  Let $Y_1, \\dots, Y_n$ be independent observations such that $\\mathbb{E}(Y_i) = 0$ and $a_i \\leq Y_i \\leq b_i$.  Let $\\epsilon > 0$.  Then, for any $t > 0$,\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\sum_{i=1}^n Y_i \\geq \\epsilon \\right) \\leq e^{-t\\epsilon} \\prod_{i=1}^n e^{t^2(b_i - a_i)^2 / 8} $$"
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   "source": [
    "**Theorem 5.5**.  Let $X_1, \\dots, X_n \\sim \\text{Bernoulli}(p)$.  Then, for any $\\epsilon > 0$,\n",
    "\n",
    "$$ \\mathbb{P}(|\\overline{X}_n - p| > \\epsilon) \\leq 2e^{-2n\\epsilon^2} $$"
   ]
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   "source": [
    "Hoeffding's inequality gives us a simple way go create a **confidence interval** for a binomial parameter $p$.  We will discuss confidence intervals later but here is the basic idea.  Let $\\alpha > 0$ and let\n",
    "\n",
    "$$ \\epsilon_n = \\left\\{ \\frac{1}{2n} \\log \\left( \\frac{2}{\\alpha} \\right) \\right\\}^{1/2} $$\n",
    "\n",
    "By Hoeffding's inequality,\n",
    "\n",
    "$$ \\mathbb{P}(|\\overline{X}_n - p| > \\epsilon_n) \\leq 2e^{-2n\\epsilon_n^2} = \\alpha $$\n",
    "\n",
    "Let $C = (\\overline{X}_n - \\epsilon, \\overline{X}_n + \\epsilon)$. Then, $\\mathbb{P}(\\text{not } C \\in p) = \\mathbb{P}(|\\overline{X}_n - p| > \\epsilon) \\leq \\alpha$.  Hence, $\\mathbb{P}(p \\in C) \\geq 1 - \\alpha$, that is, the random interval $C$ traps the true parameter $p$ with probability $1 - \\alpha$; we call $C$ a $1 - \\alpha$ confidence interval.  More on this later."
   ]
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   "source": [
    "### 5.3 Cauchy-Schwartz and Jensen Inequalities"
   ]
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   "source": [
    "This section contains two inequalities on expected values that are often useful."
   ]
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   "source": [
    "**Theorem 5.7 (Cauchy-Schwartz Inequalities)**.  If $X$ and $Y$ have finite variances then\n",
    "\n",
    "$$ \\mathbb{E}|XY| \\leq \\sqrt{\\mathbb{E}\\left(X^2\\right) \\mathbb{E}\\left(Y^2\\right)} $$"
   ]
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   "source": [
    "Recall that a function $g$ is **convex** if for each $x, y$ and each $\\alpha \\in [0, 1]$,\n",
    "\n",
    "$$ g(\\alpha x + (1 - \\alpha)y) \\leq \\alpha g(x) + (1 - \\alpha) g(y) $$\n",
    "\n",
    "If $g$ is twice differentiable, then the convexity reduces to checking that $g''(x) \\geq 0$ for all $x$.  It can be shown that if $g$ is convex then it lies above any line that touches $g$ at some point, called a tangent line.  A function $g$ is **concave** if $-g$ is convex.  Examples of convex functions are $g(x) = -x^2$ and $g(x) = \\log x$."
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   "source": [
    "**Theorem 5.8 (Jensen's Inequality)**.  If $g$ is convex then\n",
    "\n",
    "$$ \\mathbb{E}g(X) \\geq g(\\mathbb{E}X) $$\n",
    "\n",
    "If $g$ is concave then\n",
    "\n",
    "$$ \\mathbb{E}g(X) \\leq g(\\mathbb{E}X) $$"
   ]
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   "source": [
    "**Proof**.  Let $L(x) = a + bx$ be a line, tangent to the $g(x)$ at the point $\\mathbb{E}(X)$.  Since $g$ is convex, it lies above the line $L(x)$. So,\n",
    "\n",
    "$$ \\mathbb{E}g(X) \\geq \\mathbb{E}L(X) = \\mathbb{E}(a + bX) = a + b\\mathbb{E}(X) = L(\\mathbb{E}(X)) = g(\\mathbb{E}(X))$$"
   ]
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   "source": [
    "From Jensen's inequality we see that $\\mathbb{E}X^2 \\geq (\\mathbb{E}X)^2$ and $\\mathbb{E}(1/X) \\geq 1 / \\mathbb{E}(X)$.  Since log is concave, $\\mathbb{E}(\\log X) \\leq \\log \\mathbb{E}(X)$. For example, suppose that $X \\sim N(3, 1)$.  Then $\\mathbb{E}(1 / X) \\geq 1/3$."
   ]
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   "source": [
    "### 5.4 Technical Appendix: Proof of Hoeffding's Inequality"
   ]
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   "source": [
    "We will make use of the exact form of Taylor's theorem:  if $g$ is a smooth function, then there is a number $\\xi \\in (0, u)$ such that $g(u) = g(0) + u g'(0) + \\frac{u^2}{2}g''(\\xi)$."
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    "**Proof of Theorem 5.4**.  For any $t > 0$, we have, from Markov's inequality, that\n",
    "\n",
    "$$ \\mathbb{P}\\left( \\sum_{i=1}^n Y_i \\geq \\epsilon \\right) = \\mathbb{P}\\left( t \\sum_{i=1}^n Y_i \\geq t \\epsilon \\right)\n",
    "= \\mathbb{P}\\left( e^{t \\sum_{i=1}^n Y_i} \\geq e^{t \\epsilon} \\right) \\leq e^{-t\\epsilon} \\mathbb{E}\\left( e^{t \\sum_{i=1}^n Y_i}\\right) = e^{-t\\epsilon} \\prod_i \\mathbb{E}\\left(e^{tY_i}\\right) $$\n",
    "\n",
    "Since $a_i \\leq Y_i \\leq b_i$, we can write $Y_i$ as a convex combination of $a_i$ and $b_i$, namely, $Y_i = \\alpha b_i + (1 - \\alpha) a_i$ where $\\alpha = (Y_i - a_i) / (b_i - a_i)$.  So, by the convexity of $e^{ty}$ we have\n",
    "\n",
    "$$ e^{tY_i} \\leq \\frac{Y_i - a_i}{b_i - a_i} e^{tb_i} + \\frac{b_i - Y_i}{b_i - a_i} e^{ta_i} $$\n",
    "\n",
    "Take expectations of both sides and use the fact that $\\mathbb{E}(Y_i) = 0$ to get\n",
    "\n",
    "$$ \\mathbb{E}e^{tY_i} \\leq - \\frac{a_i}{b_i - a_i} e^{tb_i} + \\frac{b_i}{b_i - a_i} e^{ta_i} = e^{g(u)} $$\n",
    "\n",
    "where $u = t(b_i - a_i)$, $g(u) = -\\gamma u + \\log (1 - \\gamma + \\gamma e^u)$ and $\\gamma = -a_i / (b_i - a_i)$.\n",
    "\n",
    "Note that $g(0) = g'(0) = 0$.  Also, $g''(u) \\leq 1/4$ for all $u > 0$.  By Taylor's theorem, there is a $\\xi \\in (0, u)$ such that\n",
    "\n",
    "$$ g(u) = g(0) + u g'(0) + \\frac{u^2}{2} g(\\xi) = \\frac{u^2}{2} g(\\xi) \\leq \\frac{u^2}{8} = \\frac{t^2(b_i - a_i)^2}{8} $$\n",
    "\n",
    "Hence,\n",
    "\n",
    "$$ \\mathbb{E}e^{tY_i} \\leq e^{g(u)} \\leq e^{t^2(b_i - a_i)^2/8} $$\n",
    "\n",
    "and the result follows."
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   "source": [
    "**Proof of Theorem 5.5**.  Let $Y_i = (1 / n)(X_i - p)$.  Then $\\mathbb{E}(Y_i) = 0$ and $a \\leq Y_i \\leq b$ where $a = -p/n$ and $b = (1 - p) / n$.  Also, $(b - a)^2 = 1/n^2$.  Applying Theorem 5.4 we get\n",
    "\n",
    "$$ \\mathbb{P}\\left(\\overline{X}_n - p > \\epsilon\\right) = \\mathbb{P}\\left( \\sum_i Y_i > \\epsilon \\right) \\leq e^{-t\\epsilon} e^{t^2/(8n)}$$\n",
    "\n",
    "The above holds for any $t > 0$.  In particular, take $t = 4n\\epsilon$ and we get $\\mathbb{P}\\left(\\overline{X}_n - p > \\epsilon\\right)  \\leq e^{-2n\\epsilon^2}$.  By a similar argument we can show that $\\mathbb{P}\\left(\\overline{X}_n - p < \\epsilon\\right)  \\leq e^{-2n\\epsilon^2}$.  Putting those together we get $\\mathbb{P}\\left(|\\overline{X}_n - p| >  \\epsilon\\right)  \\leq 2e^{-2n\\epsilon^2}$."
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   "source": [
    "### 5.6 Exercises"
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   "source": [
    "**Exercise 5.6.1**.  Let $X \\sim \\text{Exponential}(\\beta)$.  Find $\\mathbb{P}(|X - \\mu_X| > k \\sigma_X)$ for $k > 1$.  Compare this to the bound you get from Chebyshev's inequality."
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    "**Solution**.\n",
    "\n",
    "Let $F$ be the CDF of $X$.  We have:\n",
    "\n",
    "$$ \n",
    "\\begin{align}\n",
    "\\mathbb{P}(|X - \\mu_X| > k \\sigma_X) &= 1 - \\mathbb{P}(-k \\sigma_X < X - \\mu_X < k \\sigma_X) \\\\\n",
    "&= 1 - \\mathbb{P}(\\mu_X - k \\sigma_X < X < \\mu_X + k \\sigma_X) \\\\\n",
    "&= 1 - F(\\mu_X + k \\sigma_X) + F(\\mu_X - k \\sigma_X) \\\\\n",
    "&= 1 - 1 + \\exp\\left\\{ -\\frac{\\left(\\beta + k \\beta\\right)^+}{\\beta} \\right\\} + 1 - \\exp\\left\\{-\\frac{\\left(\\beta - k \\beta\\right)^+}{\\beta} \\right\\} \\\\\n",
    "&= 1 + e^{-(1+k)^+ } - e^{-(1-k)^+} \n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "where $(a)^+ = \\max \\{ a, 0 \\}$.\n",
    "\n",
    "On the other hand, Chebyshev's bound provides, for $t = k\\sigma_X$,\n",
    "\n",
    "$$ \\mathbb{P}(|X - \\mu_X| \\geq k \\sigma_X) \\leq \\frac{\\sigma_X^2}{k^2 \\sigma_X^2}  = \\frac{1}{k^2} $$\n",
    "\n",
    "which is a weaker bound."
   ]
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "def f(k):\n",
    "    return 1 + np.exp(-np.maximum(1+k, 0)) - np.exp(-np.maximum(1 - k, 0))\n",
    "\n",
    "def chebyshev(k):\n",
    "    # Limit upper bound to 1, since probability is always under 1\n",
    "    return np.minimum(1 / (k**2), 1)\n",
    "\n",
    "kk = np.arange(0.01, 10, step = 0.01)\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "\n",
    "plt.plot(kk, f(kk), label='Calculated probability')\n",
    "plt.plot(kk, chebyshev(kk), label='min(Chebyshev bound, 1)')\n",
    "plt.yscale('log')\n",
    "plt.xlabel('k')\n",
    "plt.ylabel('Probability bound')\n",
    "plt.legend(loc='lower left')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 5.6.2**.  Let $X \\sim \\text{Poisson}(\\lambda)$.  Use Chebyshev's inequality to show that $\\mathbb{P}(X \\geq 2\\lambda) \\leq 1 / \\lambda$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  We have $\\mu_X = \\lambda$ and $\\sigma_X^2 = \\lambda$, so Chebyshev's gives us:\n",
    "\n",
    "$$ \\mathbb{P}(|X - \\lambda| \\geq t) \\leq \\frac{\\lambda}{t^2} $$\n",
    "\n",
    "If we make $t = \\lambda$, we get\n",
    "\n",
    "$$ \\mathbb{P}(X \\geq 2\\lambda) = \\mathbb{P}(|X - \\lambda| \\geq \\lambda) \\leq \\frac{1}{\\lambda} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 5.6.3**.  Let $X_1, \\dots, X_n \\sim \\text{Bernoulli}(p)$ and $\\overline{X}_n = n^{-1} \\sum_{i=1}^n X_i$.  Bound $\\mathbb{P}(|\\overline{X}_n - p| > \\epsilon)$ using Chebyshev's inequality and using Hoeffding's inequality.\n",
    "\n",
    "Show that, when $n$ is large, the bound from Hoeffding's inequality is smaller than the bound from Chebyshev's inequality."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.  Note that $\\mathbb{E}(\\overline{X}_n) = p$ and $\\mathbb{V}(\\overline{X}_n) = p(1-p) / n$, since $n \\overline{X}_n \\sim \\text{Binomial}(n, p)$.\n",
    "\n",
    "Using Chebyshev's inequality,\n",
    "\n",
    "$$ \\mathbb{P}(|\\overline{X}_n - p| \\geq \\epsilon) \\leq \\frac{p(1 - p)}{n \\epsilon^2} $$\n",
    "\n",
    "Using Hoeffding's inequality,\n",
    "\n",
    "$$ \\mathbb{P}(|\\overline{X}_n - p| > \\epsilon) \\leq 2e^{-2n\\epsilon^2} $$\n",
    "\n",
    "The bound provided by Hoeffding's inequality is $O(e^{-2n\\epsilon^2})$, while the bound provided by Chebyshev's inequality is $O(n^{-1})$, therefore the bound from Hoeffding's inequality is smaller for a sufficiently large $n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 5.6.4**.  Let $X_1, \\dots, X_n \\sim \\text{Bernoulli}(p)$.\n",
    "\n",
    "**(a)**  Let $\\alpha > 0$ be fixed and define\n",
    "\n",
    "$$ \\epsilon_n = \\sqrt{\\frac{1}{2n} \\log \\left( \\frac{2}{\\alpha}\\right)} $$\n",
    "\n",
    "Let $\\hat{p}_n = n^{-1} \\sum_{i=1}^n X_i$.  Define $C_n = (\\hat{p}_n - \\epsilon_n, \\hat{p}_n + \\epsilon_n)$.  Use Hoeffding's inequality to show that\n",
    "\n",
    "$$ \\mathbb{P}(p \\in C_n) \\geq 1 - \\alpha $$\n",
    "\n",
    "We call $C_n$ a *$1 - \\alpha$ confidence interval for $p$*. In practice, we truncate the interval so it does not go below 0 or above 1.\n",
    "\n",
    "**(b) (Computer Experiment)** Let's examine the properties of this confidence interval.  Let $\\alpha = 0.05$ and $p = 0.4$.  Conduct a simulation study to see how often the interval contains $p$ (called the coverage).  Do this for various values of $n$ between 1 and 10000.  Plot the coverage versus $n$.\n",
    "\n",
    "**(c)** Plot the length of the interval versus $n$.  Suppose we want the length of the interval to be no more than .05.  How large should $n$ be?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**.\n",
    "\n",
    "**(a)**  The result is immediate from replacing $\\epsilon_n$ into Hoeffding's inequality,\n",
    "\n",
    "$$ \\mathbb{P}(|\\hat{p}_n - p| > \\epsilon_n) \\leq 2e^{-2n\\epsilon_n^2} = \\alpha $$\n",
    "\n",
    "since $\\mathbb{E}(\\hat{p}_n) = p$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(b)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "4eb014da8e2548bc8c1fa9bc6584ca3a",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "HBox(children=(IntProgress(value=0, max=50000), HTML(value='')))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import bernoulli\n",
    "from tqdm import tqdm_notebook\n",
    "\n",
    "alpha = 0.05\n",
    "p = 0.4\n",
    "\n",
    "B = 50000\n",
    "N = 10000\n",
    "\n",
    "nn = np.arange(1, N + 1)\n",
    "epsilon_n = np.sqrt((1 / (2 * nn)) * np.log(2 / alpha))\n",
    "\n",
    "p_hat = np.empty((B, N))\n",
    "for i in tqdm_notebook(range(B)):\n",
    "    X = bernoulli.rvs(p, size=N, random_state=i)\n",
    "    p_hat[i] = np.cumsum(X) / nn\n",
    "\n",
    "coverage = np.mean((p_hat + epsilon_n >= p) & (p_hat - epsilon_n <= p), axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(12, 8))\n",
    "plt.plot(nn, coverage)\n",
    "plt.xlabel('n')\n",
    "plt.ylabel('Coverage')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(c)**  The length of the interval is $\\min \\{\\hat{p}_n + \\epsilon_n, 1 \\} - \\max \\{ \\hat{p}_n - \\epsilon_n, 0\\}$.  As $\\hat{p}_n \\rightarrow p$, let's plot the approximation on the limit case, which is just $2 \\epsilon_n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Smallest n with interval length under .05: 2952\n"
     ]
    }
   ],
   "source": [
    "plt.figure(figsize=(12, 8))\n",
    "plt.plot(nn, 2 * epsilon_n, label='Interval length')\n",
    "plt.xlabel('n')\n",
    "plt.ylabel(r'$2\\epsilon_n$')\n",
    "plt.hlines(.05, xmin=0, xmax=N, color='red', label='Threshold')\n",
    "plt.yscale('log')\n",
    "plt.legend(loc='upper right')\n",
    "plt.show()\n",
    "\n",
    "selected_n = nn[np.argmax(2 * epsilon_n <= .05)]\n",
    "print('Smallest n with interval length under .05: %i' % selected_n)"
   ]
  }
 ],
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