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diff --git a/math/statistics.page b/math/statistics.page new file mode 100644 index 0000000..446450f --- /dev/null +++ b/math/statistics.page @@ -0,0 +1,80 @@ +Statistics +==================== + +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: + +$$\bar{x}= + \frac{ \sum_{i=1}^{N} x_i / \simga_{i}^2}{\sum_{i=1}^{N} 1/\simga_{i}^2} + \pm \sqrt{ \frac{1}{\sum_{i=1}^{N} 1/\simga_{i}^2}}$$ + +$\Chi^2$ 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 +bad. + |