The issue seems to be that you want the "column" to the right of the brace to be centered rather than left justified, which is what happens in all of the cases environments (at least, in all of the ones that I know about). To fix this this the OP's idea of using an array seems to be the way to go, but a few tweaks are needed.
- First, if you use
\left\{
then there has to be a matching \right<delimiter>
. As there isn't a matching right delimiter, in this case, you can use \right.
. Personally, I don't like the sizes that \left....\right
produces so, instead, I have defined a macro \Bigger
that will scale the following delimiter to the specified height in millimeters, which defaults to 7mm
. Except for the optional argument, this is similar to the \bigl
, \biggl
, \Big
... amsmath macros.
- Next, as the OP says, the rows of the array are too close together. You can adjust the spacing between rows using the optional argument to the
\\
command. So \\[3mm]
will add an extra 3mm
between the rows, which seems about right.
- The
array
environment comes with padding, so there is too much space between the brace and the equations. You can remove the left-padding by adding @{}
to the array specifications.
- I have added some punctuation.
- Finally, as suggested in the comments,
\mid
is better than |
.
With these changes the MWE produces:

This is pretty close to the "ideal" image given at the top of the question. Here is the full code:
\documentclass{article}
\usepackage{amsmath}
% an ams-style operator for resizing delimiters
% Usage: \Bigger[, \Bigger[8][, \Bigger[10](, ...
\newcommand\Bigger[2][7]{\left#2\rule{0mm}{#1truemm}\right.}
\begin{document}
\begin{displaymath}
P[\chi_{n+1}=j\mid\chi_{n}=i]=
\Bigger[14]\{\begin{array}{@{}cl}
\dfrac{N-i}{N}, & \text{si }j=i+1,\\[3mm]
\dfrac{i}{N}, & \text{si }j=i-1,\\[3mm]
0, & \text{en otro caso}.
\end{array}
\end{displaymath}
\end{document}
\mid
instead of|
to denote conditioning on some event.\newcommand{\given}{\mid}
, but yes.)