@newcount@calcpageno
@newtoks@calcoldeverypar @calcoldeverypar=@everypar
@everypar={@calceverypar@the@calcoldeverypar}
-@ifx@turnoffactive@undefinedzzz@def@turnoffactive{}@fi
@ifx@ninett@undefinedzzz@font@ninett=cmtt9@fi
@catcode`@\=0 \catcode`\@=11
\r@ggedbottomtrue
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
\afterdisplay
@end group
@end ifnottex
@tex
-\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
@end group
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
\times
@end group
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \eqalign{ x &+ a y = 6 \cr
x &+ b y = 10}
@samp{trn(A)*A*X = trn(A)*B}.
@end ifnottex
@tex
-\turnoffactive
$A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
@end tex
Now
@end group
@end ifnottex
@tex
-\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ m = {N \sum x y - \sum x \sum y \over
N \sum x^2 - \left( \sum x \right)^2} $$
@samp{sum(x y)}.)
@end ifnottex
@tex
-\turnoffactive
These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
$\sum x y$.)
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ b = {\sum y - m \sum x \over N} $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \displaylines{
\qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
+ f(a+(n-2)h) + f(a+(n-1)h)) $$
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \eqalign{ x &+ a y = 6 \cr
x &+ b y = 10}
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ m \times x + b \times 1 = y $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ 3 (3 a + b - 511 m) + c - 511 n $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ 9 a + 3 b + c - 511\times3 m - 511 n $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ 9 a + 3 b + c - 511 n^{\prime} $$
\afterdisplay
@end group
@end example
@tex
-\turnoffactive
$$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
@end tex
@sp 1
@end group
@end example
@tex
-\turnoffactive
$$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
@end tex
@sp 2
@end group
@end example
@tex
-\turnoffactive
$$ 2 + 3 \to 5 $$
$$ 5 $$
@end tex
@end group
@end example
@tex
-\turnoffactive
$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
{\let\to\Rightarrow
$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
@end group
@end example
@tex
-\turnoffactive
$$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
$$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
@end tex
@end example
@end ifnottex
@tex
-\turnoffactive
$$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
$$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
$$ \code{fvl}(r, n, p) = p (1 + r)^n $$
and @kbd{H I f G} [@code{gammaG}] commands.
@end ifnottex
@tex
-\turnoffactive
The functions corresponding to the integrals that define $P(a,x)$
and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
@texline @math{1 /\sigma^2}.
@infoline @expr{1 / s^2}.
@tex
-\turnoffactive
$$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
\displaystyle \sum { 1 \over \sigma_i^2 } } $$
@end tex
of the input errors. (I.e., the variance is the reciprocal of the
sum of the reciprocals of the variances.)
@tex
-\turnoffactive
$$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
@end tex
If the inputs are plain
then assuming each value's error is equal to this standard
deviation.)
@tex
-\turnoffactive
$$ \sigma_\mu^2 = {\sigma^2 \over N} $$
@end tex
defined as the reciprocal of the arithmetic mean of the reciprocals
of the values.
@tex
-\turnoffactive
$$ { N \over \displaystyle \sum {1 \over x_i} } $$
@end tex
equal to the @code{exp} of the arithmetic mean of the logarithms
of the data values.
@tex
-\turnoffactive
$$ \exp \left ( \sum { \ln x_i } \right ) =
\left ( \prod { x_i } \right)^{1 / N} $$
@end tex
replacing the two numbers with their arithmetic mean and geometric
mean, then repeating until the two values converge.
@tex
-\turnoffactive
$$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
@end tex
the differences between the values and the mean of the @expr{N} values,
divided by @expr{N-1}.
@tex
-\turnoffactive
$$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
@end tex
data values, so that the mean computed from the input is itself
only an estimate of the true mean.
@tex
-\turnoffactive
$$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
@end tex
is taken as the square root of the sum of the squares of the two
input errors.
@tex
-\turnoffactive
$$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
$$ \sigma_{x\!y}^2 =
{\displaystyle {1 \over N-1}
product of their standard deviations. (There is no difference
between sample or population statistics here.)
@tex
-\turnoffactive
$$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
@end tex
@end example
@end ifnottex
@tex
-\turnoffactive
-\turnoffactive
\beforedisplay
$$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
5 & 7 & 9 & 11 & 13 }
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
\afterdisplay
the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}}
produces the result 55.
@tex
-\turnoffactive
$$ \sum_{k=1}^5 k^2 = 55 $$
@end tex
projects / emacs.git / commitdiff