Determine Parameters of gaussian distribution to fit a certain Image-Filter
The filter you referenced is known as a Binomial filter. It is an approximation to a Gaussian, but for smaller filters it's a very crude one. The design is oriented more toward efficiency than accuracy. Rather than sampling a Gaussian directly, the idea behind the approximation is based on the Central Limit Theorem. In this case, it means that a small moving average filter $[ 1 \space  1 ]$ convolved with itself over and over will become more and more Gaussian. It also happens to yield the rows of Pascal's Triangle. The coefficients for larger filters are given by the following formula:

$$
{N \choose k} = \frac{N!}{(N - n)!\space N!}.
$$

For example, for $N = 5$, taking the outer product of the corresponding row of Pascal's triangle with itself yields

$$
\begin{bmatrix} 1 \\ 4 \\ 6 \\ 4 \\ 1 \end{bmatrix}\begin{bmatrix} 1 &amp; 4 &amp; 6 &amp; 4 &amp; 1 \end{bmatrix} = 
\begin{bmatrix} 
1 &amp;   4 &amp;   6  &amp;  4  &amp;  1 \\
4 &amp;  16 &amp;  24 &amp;  16  &amp;  4 \\
6 &amp;  24 &amp;  36 &amp;  24  &amp;  6 \\
4 &amp;  16 &amp;  24 &amp;  16  &amp;  4 \\
1 &amp;   4 &amp;   6 &amp;   4  &amp;  1 
\end{bmatrix}.
$$

The paper I referenced in the first link has the following diagram which shows how the quality of the approximation increases with increasing filter order: