Bogleheads:Sandbox

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$$ \operatorname{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}} $$

The average return for this four-year period is calculated as the arithmetic mean of the annual returns. The arithmetic mean often is referred to simply as the mean. In finance theory this is commonly referred to as the expected value of the returns, which here is denoted as E(r). Here is the calculation of the mean, or expected value, for our example (for simplicity, the percent signs are dropped):

$$ E(r) = \frac{(-36.55) + 25.94 + 14.82 + 2.07}{4} = 1.57 $$

So the mean, or expected value, of the annual returns is 1.57%.

Note an alternative way to express the calculation (that will be useful later):

$$ E(r) = \frac{1}{4} \times (-36.55) + \frac{1}{4} \times 25.94 + \frac{1}{4} \times 14.82 + \frac{1}{4} \times 2.07 = 1.57 $$

The average of the annual returns is useful information, but it doesn't indicate anything about the variation (dispersion) of returns.

From Wikipedia
(Compare to Error function - Wikipedia, the free encyclopedia)

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as:


 * $$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt.$$

(When x is negative, the integral is interpreted as the negative of the integral from x to zero.)

The complementary error function, denoted erfc, is defined as


 * $$\begin{align}

\operatorname{erfc}(x) & = 1-\operatorname{erf}(x) \\ & = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt. \end{align}$$

The imaginary error function, denoted erfi, is defined as


 * $$\operatorname{erfi}(z) = -i\,\,\operatorname{erf}(i\,z)$$

The complex error function, denoted w(x) and also known as the Faddeeva function, is defined as


 * $$w(x) = e^{-x^2}{\operatorname{erfc}}(-ix) = e^{-x^2}[1+i\,\,\operatorname{erfi}(x)]$$

The name "error function"
The error function is used in measurement theory (using probability and statistics), and although its use in other branches of mathematics has nothing to do with the characterization of measurement errors, the name has stuck.

The error function is related to the cumulative distribution $$\Phi$$, the integral of the standard normal distribution (the "bell curve"), by


 * $$\Phi (x) = \frac{1}{2}+ \frac{1}{2} \operatorname{erf} \left(x/ \sqrt{2}\right)$$

The error function, evaluated at $$\frac{x}{\sigma \sqrt{2}}$$ for positive x values, gives the probability that a measurement, under the influence of normally distributed errors with standard deviation $$\sigma$$, has a distance less than x from the mean value. This function is used in statistics to predict behavior of any sample with respect to the population mean. This usage is similar to the Q-function, which in fact can be written in terms of the error function.

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