Fixed section headers

Lab3-Freq.ipynb

@@ -196,7 +196,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###What is data?\n",
+    "### What is data?\n",
     "\n",
     "In labs before, you have seen datasets. As so in the example above. You have seen probability distributions of this data. Calculated means. Calculated standard deviations."
    ]
@@ -205,7 +205,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "####Pandas code for the week\n",
+    "#### Pandas code for the week\n",
     "\n",
     "We'll keep showing some different aspects of Pandas+Seaborn each week. For example, you can very easily calculate correlations"
    ]
@@ -327,7 +327,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "####Samples vs population\n",
+    "#### Samples vs population\n",
     "\n",
     "But we have never aked ourselves the philosophical question: what is data? **Frequentist statistics** is one answer to this philosophical question. It treats data as a **sample** from an existing **population**.\n",
     "\n",
@@ -338,12 +338,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###Choosing a model\n",
+    "### Choosing a model\n",
     "\n",
     "Let us characterize our particular sample statistically then, using a *probability distribution*\n",
     "\n",
     "\n",
-    "####The Exponential Distribution\n",
+    "#### The Exponential Distribution\n",
     "\n",
     "The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous Poisson process.\n",
     "\n",
@@ -447,7 +447,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###How would we draw from this distribution?\n",
+    "### How would we draw from this distribution?\n",
     "\n",
     "Lets use the built in machinery in `scipy.stats`:"
    ]
@@ -623,7 +623,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "####An aside: The Poisson Distribution\n",
+    "#### An aside: The Poisson Distribution\n",
     "\n",
     "The *Poisson Distribution* is defined for all positive integers: \n",
     "\n",
@@ -834,7 +834,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###Segue: many samples on the binomial"
+    "### Segue: many samples on the binomial"
    ]
   },
   {
@@ -979,7 +979,7 @@
     "\n",
     "We do it here for the mean of the time differences. We could also do it for its inverse, $\\lambda$.\n",
     "\n",
-    "####Non Parametric bootstrap\n",
+    "#### Non Parametric bootstrap\n",
     "\n",
     "Resample the data! We can then plot the distribution of the mean time-difference."
    ]
@@ -1028,7 +1028,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "####Parametric Bootstrap\n",
+    "#### Parametric Bootstrap\n",
     "\n",
     "And here we do it in a parametric way. We get an \"estimate\" of the parameter from our sample, and them use the exponential distribution to generate many datasets, and then fir the parameter on each one of those datasets. We can then plot the distribution of the mean time-difference."
    ]

Lab3-Stats.ipynb

@@ -4,7 +4,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "#Sampling and Distributions"
+    "# Sampling and Distributions"
    ]
   },
   {
@@ -41,7 +41,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###Expectations and Variance\n",
+    "### Expectations and Variance\n",
     "\n",
     "The **expectation value** of a quantity with respect to the a distribution is the weighted sum of the quantity where the weights are probabilties from the distribution. For example, for the random variable $X$:\n",
     "\n",
@@ -64,7 +64,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###The Law of Large Numbers\n",
+    "### The Law of Large Numbers\n",
     "\n",
     "Lets keep increasing the length of the sequence of coin flips n, and compute a running average $S_n$ of the coin-flip random variables,\n",
     "$$S_n = \\frac{1}{n} \\sum_{i=1}^{n} x_i .$$\n",
@@ -511,7 +511,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###The Gaussian Distribution\n",
+    "### The Gaussian Distribution\n",
     "\n",
     "We saw in the last section that the sampling distribution of the mean itself has a mean $\\mu$ and variance $\\frac{\\sigma^2}{N}$. This distribution is called the **Gaussian** or **Normal Distribution**, and is probably the most important distribution in all of statistics.\n",
     "\n",
@@ -577,7 +577,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "###The Central Limit Theorem\n",
+    "### The Central Limit Theorem\n",
     "\n",
     "The reason for the distribution's importance is the Central Limit Theorem(CLT). The theorem is stated as thus, very similar to the law of large numbers:\n",
     "\n",
@@ -616,7 +616,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "####An application to elections: Binomial distribution in the large n, large k limit\n",
+    "#### An application to elections: Binomial distribution in the large n, large k limit\n",
     "For example, consider the binomial distribution Binomial(n,k, p) in the limit of large n. The number of successes k in n trials can be ragarded as the sum of n IID Bernoulli variables with values 1 or 0. Obviously this is applicable to a large sequence of coin tosses, or to the binomial sampling issue that we encountered earlier in the case of the polling. \n",
     "\n",
     "Using the CLT we can replace the binomial distribution at large n by a gaussian where k is now a continuous variable, and whose mean is the mean of the binomial $np$ and whose variance is $np(1-p)$, since\n",