I have a study area (forest) around 25he. I would like to randomly have sampling points (center of the sampling plots) there. How many points (plots) I need in related to the area of my case study? is there any rule for calculating the number of the random points?
Do you have a specific hypothesis? What type of analysis or statistical model are you intending? The number and spatial distribution of sample points is completely dependent on these considerations.
Commonly, random sampling is applied to avoid spatial autocorrelation, which can have a profound influence on certain linear models. However, if you are wanting to incorporate spatial process into a model, you need to ensure that the sample captures some type of autocorrelation process in the data, which can sometimes be accomplished by simply increasing the sample intensity. In the case of sampling for a spatial process I would recommend a stratified random sample in lieu of a purely-random sample.
There are two important considerations:
1) Are you capturing or minimizing autocorrelation - In sampling autocorrelation this could mean creating a sample that represents a specific type of autocorrelation following a hypotheses of spatial process or just rejecting a null of CSR (Complete Spatial Randomness). To minimize the effect of autocorrelation you could, in theory, generate a random sample that accepts a CSR Poisson process.
2) Are you adequately capturing statistical variation - This has to do with sampling the distributions of your dependent and independent variables. For instance, it you want to make an estimate the incorporates elevation [500-2500] but you only sampled in the range of [800-1000] then the sample variation of elevation has not been adequately captured. To evaluate this just examine the sample distributions against the population distributions. This is a critical step in exploratory data analysis.
If you are wanting to make a spatial estimate (eg., model forest biomass, species distribution), capturing spatial and statistical variation are both of paramount importance. In directly addressing your question of "how many samples" you could use a mean expected random distance and test multiple realizations of a random n to test if you are accepting or rejecting a random spatial process. This can be done using a simple test statistic such as the nearest neighbor index.
Mean Nearest Neighbor Distance (observed); D(nn) = sum(min(Dij)/N) Mean Random Distance (expected); D(e) = 0.5 SQRT(A/N) Nearest Neighbor Index; NNI = D(nn)/D(e) Where; D = neighbor distance, A = Area
If you are working in the ArcGIS ecosystem, this statistic can be calculated using the Average Nearest Neighbor tool in the Spatial Statistics Toolbox. This is one of the few tools in the toolbox that I recommend using but keep in mind that it is a very simple measure of spatial process. The advantage is that it works entirely on the point locations, unlike statistics like Moran's-I or Getis-Ord that require that the points have an associated continuous value.
Please keep in mind that sample design is not a trivial issue and more consideration than "what script should I use" should be given to the design. I spend considerable time with students performing post mortems on their sample design and how it cannot address their question/analysis. In my quantitative ecology course I spend over two weeks on sample/experimental design alone.