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Roger Labbe 2016-01-09 08:51:24 -08:00
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03-Gaussians.ipynb
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"cell_type": "markdown",
"metadata": {},
"source": [
"So what does this curve *mean*? Assume we have a thermometer which reads 22$\\,^{\\circ}C$. No thermometer is perfectly accurate, and so we normally expect that thermometer will read slightly plus or minus that temperature each time we read it. However, a theorem called *Central Limit Theorem* states that if we make many measurements that the measurements will be normally distributed. So, when we look at this chart we can \"sort of\" think of it as representing the probability of the thermometer reading a particular value given the actual temperature of 22$^{\\circ}C$. \n",
"So what does this curve *mean*? Assume we have a thermometer which reads 22°C. No thermometer is perfectly accurate, and so we normally expect that thermometer will read slightly plus or minus that temperature each time we read it. However, a theorem called *Central Limit Theorem* states that if we make many measurements that the measurements will be normally distributed. So, when we look at this chart we can \"sort of\" think of it as representing the probability of the thermometer reading a particular value given the actual temperature of 22°C. \n",
"\n",
"Recall that a Gaussian distribution is *continuous*. Think of an infinitely long straight line - what is the probability that a point you pick randomly is at 2. Clearly 0%, as there is an infinite number of choices to choose from. The same is true for normal distributions; in the graph above the probability of being *exactly* 22$^{\\circ}C$ is 0% because there are an infinite number of values the reading can take.\n",
"Recall that a Gaussian distribution is *continuous*. Think of an infinitely long straight line - what is the probability that a point you pick randomly is at 2. Clearly 0%, as there is an infinite number of choices to choose from. The same is true for normal distributions; in the graph above the probability of being *exactly* 2°C is 0% because there are an infinite number of values the reading can take.\n",
"\n",
"So what then is this curve? It is something we call the *probability density function.* The area under the curve at any region gives you the probability of those values. So, for example, if you compute the area under the curve between 20 and 22 the resulting area will be the probability of the temperature reading being between those two temperatures. \n",
"\n",
"We can think of this in Bayesian terms or frequentist terms. As a Bayesian, if the thermometer reads exactly 22$\\,^{\\circ}C$, then our belief is described by the curve - our belief that the actual (system) temperature is near 22 is very high, and our belief that the actual temperature is near 18 is very low. As a frequentist we would say that if we took 1 billion temperature measurements of a system at exactly 22$\\,^{\\circ}C$, then a histogram of the measurements would look like this curve. \n"
"We can think of this in Bayesian terms or frequentist terms. As a Bayesian, if the thermometer reads exactly 22°C, then our belief is described by the curve - our belief that the actual (system) temperature is near 22 is very high, and our belief that the actual temperature is near 18 is very low. As a frequentist we would say that if we took 1 billion temperature measurements of a system at exactly 22°C, then a histogram of the measurements would look like this curve. \n"
]
},
{