Fixed erroroneous statement about PDFs.
For some reason I stated the PDF displayed the probability, not the probability density.
This commit is contained in:
Roger Labbe committed
1 parent
545a4650a1
commit
22375983c2
1 file changed
+44
-14
@@ -1,7 +1,7 @@
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{
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"metadata": {
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"name": "",
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"signature": "sha256:71e5314e7d6889b41a67a2adfc7f9a248d967c2d98a94f399a428d8a11833da2"
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"signature": "sha256:2e309940166b97d5add0292b35ffd44bf21b1fc4427c233aed7521269e0805b1"
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},
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"nbformat": 3,
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"nbformat_minor": 0,
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@@ -43,6 +43,8 @@
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"html": [
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"<style>\n",
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"@import url('http://fonts.googleapis.com/css?family=Source+Code+Pro');\n",
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"@import url('http://fonts.googleapis.com/css?family=Vollkorn');\n",
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"@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
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"\n",
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" div.cell{\n",
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" width: 850px;\n",
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@@ -129,9 +131,9 @@
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" white-space: nowrap;\n",
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" }\n",
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" div.text_cell_render{\n",
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" font-family: 'Open sans',verdana,arial,sans-serif;\n",
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" font-family: 'Arimo',verdana,arial,sans-serif;\n",
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" line-height: 135%;\n",
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" font-size: 110%;\n",
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" font-size: 125%;\n",
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" width:750px;\n",
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" margin-left:auto;\n",
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" margin-right:auto;\n",
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@@ -141,12 +143,12 @@
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" div.output_subarea.output_text.output_pyout {\n",
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" overflow-x: auto;\n",
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" overflow-y: scroll;\n",
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" max-height: 300px;\n",
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" max-height: 50000px;\n",
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" }\n",
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" div.output_subarea.output_stream.output_stdout.output_text {\n",
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" overflow-x: auto;\n",
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" overflow-y: scroll;\n",
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" max-height: 300px;\n",
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" max-height: 50000px;\n",
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" }\n",
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" code{\n",
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" font-size: 70%;\n",
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@@ -257,13 +259,13 @@
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],
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"metadata": {},
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"output_type": "pyout",
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"prompt_number": 3,
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"prompt_number": 2,
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"text": [
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"<IPython.core.display.HTML at 0x7fdfe851eb00>"
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"<IPython.core.display.HTML at 0x7f99b0d79550>"
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]
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}
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],
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"prompt_number": 3
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"prompt_number": 2
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},
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{
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"cell_type": "heading",
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@@ -290,7 +292,7 @@
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"input": [
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"from stats import plot_gaussian\n",
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"\n",
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"plot_gaussian(mean=100, variance=15*15, xlabel='IQ')"
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"plot_gaussian(mean=100, variance=15**2, xlabel='IQ')"
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],
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"language": "python",
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"metadata": {},
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@@ -300,11 +302,11 @@
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"output_type": "display_data",
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"png": "iVBORw0KGgoAAAANSUhEUgAAAtIAAAGJCAYAAACny9QDAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
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"text": [
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"<matplotlib.figure.Figure at 0x7fdfc797ca90>"
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"<matplotlib.figure.Figure at 0x7f999856b978>"
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]
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}
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],
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"prompt_number": 4
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"prompt_number": 3
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},
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{
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"cell_type": "markdown",
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@@ -329,10 +331,38 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"A bit of nomenclature before we continue - this chart depicts the probability of of a *random variable* having any value between ($-\\infty..\\infty)$. For example, for this chart the probability of the variable being 100 is roughly 2.7%, whereas the probability of it being 80 is around 1%.\n",
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"A bit of nomenclature before we continue - this chart depicts the *probability density* of of a *random variable* having any value between ($-\\infty..\\infty)$. What does that mean? Imagine we take an infinite number of infinitely precise measurements of the speed of automobiles on a section of highway. We could then plot the results by showing the relative number of cars going at any given speed. If the average was 120 kph, it might look like this:"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"plot_gaussian(mean=120, variance=17**2, xlabel='speed(kph)')"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"metadata": {},
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"output_type": "display_data",
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"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f9994166198>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 5
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The y-axis depicts the *probability density* - the relative amount of cars that are going the speed at the corresponding x-axis.\n",
|
||||
"\n",
|
||||
"> *Random variable* will be precisely defined later. For now just think of it as a variable that can 'freely' and 'randomly' vary. A dog's position in a hallway, air temperature, and a drone's height above the ground are all random variables. The position of the North Pole is not, nor is a sin wave (a sin wave is anything but 'free').\n",
|
||||
"\n",
|
||||
"You may object that human IQs cannot be less than zero, let alone $-\\infty$. This is true, but this is a common limitation of mathematical modeling. \"The map is not the territory\" is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above very closely models the distribution of IQ test results, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. \n",
|
||||
"You may object that human IQs or automobile speeds cannot be less than zero, let alone $-\\infty$. This is true, but this is a common limitation of mathematical modeling. \"The map is not the territory\" is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above very closely models the distribution of IQ test results, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. \n",
|
||||
"\n",
|
||||
"You will see these distributions called *Gaussian distributions*, *normal distributions*, and *bell curves*. Bell curve is ambiguous because there are other distributions which also look bell shaped but are not Gaussian distributions, so we will not use it further in this book. But *Gaussian* and *normal* both mean the same thing, and are used interchangeably. I will use both throughout this book as different sources will use either term, and so I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and just talk about a *Gaussian* or *normal* - these are both typical shortcut names for the *Gaussian distribution*. "
|
||||
]
|
||||
@@ -790,7 +820,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"There are many other functions available, and if you are interested I urge you to peruse the documentation. I find the documentation to be excessively terse, but with a bit of googling you can find out what a function does and some examples of how to use it. Most of this functionality is not of immediate interest to the book, so I will leave the topic in your hands to explore. The tutorial is quite approachable, and I suggest starting there. <a href=\"http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html\">tutorial</a>[2]"
|
||||
"There are many other functions available, and if you are interested I urge you to peruse the documentation. I find the documentation to be excessively terse, but with a bit of googling you can find out what a function does and some examples of how to use it. Most of this functionality is not of immediate interest to the book, so I will leave the topic in your hands to explore. The <a href=\"http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html\">SciPy tutorial</a> [2] is quite approachable, and I suggest starting there. "
|
||||
]
|
||||
},
|
||||
{
|
||||
|
||||
Reference in new issue
Block a user