I am a research scholar in electrical engineering (power systems). I am working on probabilistic approaches for power system analysis and I am relatively new to this area. I am able to understand how to find the marginals of the wind speeds ( wind speeds at different locations will form my random vector) and correlation matrix. I am reading literature on various probabilistic approches and am curious to know if I can form a joint distribution or joint density of the random vector with the information. Please let me know your opinion or even refer to something that i can refer. I appreciate your help.
Thank You.
No you can't, except in special cases such as when the random variables are jointly Gaussian: in this case, the marginals and the pairwise correlation coefficients completely determine the distribution.
In general, the joint distribution of a random variables $X_1 \ldots X_n$ is determined by the marginal distributions $F_i(c) = \mathbb P(X_i \ge c)$ together with a function $\psi: [0,1]^n \to [0,1]$ called a copula. The copula function is given by $$\psi(c_1 \ldots c_n) = \mathbb P\left(F_1(X_1) \ge c_1, \ldots, F_n(X_n) \ge c_n\right).$$ You can read more about copulas and their uses in probability and statistics here. Regardless of what measure of correlation is used (e.g. linear correlation, rank-correlation), there are examples of different joint distributions with the same marginals and correlation.
The intuition is that the dependence structure of the variables is an infinite-dimensional object (a copula), and a one-dimensional measure of correlation will not adequately summarize it in all cases.