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authorbnewbold <bnewbold@manus.(none)>2009-11-02 10:05:48 -0800
committerbnewbold <bnewbold@manus.(none)>2009-11-02 10:05:48 -0800
commitb06b63d33ce8ac9713470463237e824a3826e749 (patch)
treed6de83b7cee7beaa244629e357ac47ae8f0172af /math/
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stats page; TODO: untested latex
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+Basic Measures
+The sample distribution has finite size and is what has been measured; the
+parent distribution is inifinite and smooth and is the limit case of the
+sample distribution.
+The mean, or average, is (of course):
+$$\langle x \rangle = \frac{1}{N} \sum_{i=1}^{N}x_i$$
+The variance is;
+$$s^{2}_x = \frac{1}{N-1}\sum^{N}_{i=1}\left(x-\langle x \rangle\right)^2$$
+The standard deviation is the square root of the variance; the standard
+deviation of the parent distribution is represented by $\sigma_x$ instead of
+$s_x$. The mean of the parent distribution is $\mu$ instead of $\bar{x}$.
+Binomial Distribution
+If we are playing a yes/no game (eg flipping a coin), the binomial distribution
+represents the probability of getting 'yes' $x$ times out of $n$ if $p$ is the
+probability of getting 'yes' for a single attempt.
+$$P(x;n,p) = \frac{n!}{x! (n-x)!} p^x (1-p)^{n-x}$$
+The mean of this distribution is $\mu = np$, and $\sigma$ = \sqrt{np (1-p)}.
+Poisson Distribution
+$$P(x,\mu) = \frac{\mu^x}{x!} e^{-\mu}$$
+The mean is $\mu$, and $\sigma=\sqrt{\mu}$.
+Gaussian Distribution
+The classic! Also called a normal distribution.
+$$P(x;\mu,\sigma) = \frac{1}{2\pi \sigma} e^{-\left(\frac{(x-\mu)^2}{2\sigma^2}\right)}$$
+The mean is $\mu$ and the deviation is $\sigma=\sqrt(\mu)$.
+Lorentzian Distribution
+This distribution represents damped resonance; it is also the Fourier
+transform of an exponentially decaying sinusoid.
+$$P(x;\mu,\Gamma) = \frac{1}{\pi} \frac{\Gamma/2}{(x-\mu)^2 + (\Gamma/2)^2}$$
+where the mean is $\mu$ and the linewidth (the width of the peak) is $\Gamma$.
+Error Analysis
+For a given measurement, the error on the mean is not the standard deviation
+(which is a measure of the statistics), it is $\frac{s_x}{\sqrt{N}}$: the
+standard deviation should stay roughly constant as $N$ gets very large, but
+the error on the mean should get smaller. More elaborately, if the errors are
+different for each individual measurement, the mean will be:
+ \frac{ \sum_{i=1}^{N} x_i / \simga_{i}^2}{\sum_{i=1}^{N} 1/\simga_{i}^2}
+ \pm \sqrt{ \frac{1}{\sum_{i=1}^{N} 1/\simga_{i}^2}}$$
+$\Chi^2$ Distribution
+$\Chi^2$ is often writen "chi-squared" and is a metric for how well a fit
+curve matches uncertain data.
+$$\Chi^2 = \sum_{i=1}^{N}\left(\frac{x_i-\mu_i}{\sigma{i}}\right)^2$$
+The number of degrees of freedom of the system is the number of measurements
+$N$ minus the number of variable parameters in a curve fit $N_c$: $\nu = N-N_c$.
+The reduced $\Chi^2$ value is $\Chi^{2}_r = \Chi^2 /\nu$. You want $\Chi^{2}_r$
+to be around (but not exactly!) 1; if it is significantly larger there are
+probably too many degrees of freedom, while if significantly smaller the fit is