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######################################################################
# On arithmetics operating on distributions in package "distr"
######################################################################
Attention:
Special caution is due in the followin issues
%--------------------------------------------------------------------
% Interpretation of arithmetics
%--------------------------------------------------------------------
Arithmetics on distribution objects are understood as operations on
corresponding random variables (r.v.'s) and _not_ on distribution
functions or densities;
e.g.
sin( Norm() + 3 * Norm() ) + 2
returns a distribution object representing the distribution of the r.v.
sin(X+3*Y)+2
where X and Y are r.v.'s i.i.d. N(0,1).
%--------------------------------------------------------------------
% Adjusting accuracy
%--------------------------------------------------------------------
Binary operators like "+", "-" would loose their elegant calling
e1 + e2 if they had to be called with an extra argument controlling
their accuracy. Therefore, this accuracy is controlled by global options.
These options are inspected and set by distroptions(), getdistrOption(),
see ?distroptions
%--------------------------------------------------------------------
% Multiple instances in expressions and independence
%--------------------------------------------------------------------
Special attention has to be paid to arithmetic expressions of
distributions involving multiple instances of the same symbol:
/-> All arising instances of distribution objects in arithmetic
expressions are assumed stochastically independent. <-/
As a consequence, whenever in an expression, the same symbol for
an object occurs more than once, every instance means a new
independent distribution.
So for a distribution object X, the expressions X+X and 2*X are _not_
equivalent.
The first means the convolution of distribution X with distribution
X, i.e. the distribution of the r.v. X1 + X2, where X1 and X2 are
identically distributed according to X.
In contrast to this, the second expression means the distribution of
the r.v. 2 X1 = X1 + X1, where again X1 is distributed according to X.
Hence always use 2*X, when you want to realize the second case.
Similar caution is due for X^2 and X*X and so on.
%--------------------------------------------------------------------
% Simulation based results varying from call to call
%--------------------------------------------------------------------
At several instances (in particular for non-monotone functions from group
Math like sin(), cos()) new distributions are generated by means of
RtoDPQ, RtoDPQ.d, RtoDPQ.LC. In these functions, slots d, p, q are
filled by simulating a large number of random variables, hence they are
stochastic estimates.
So don't be surprised if they will change from call to call.
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