To overcome the assumption of same family of marginal density function for all flood variables, the concept of copula has been introduced.
The marginal density function of the th ordered eigenvalue can be obtained by [19] (16).
Therefore, the marginal density function is obtained by summing the joint distribution over all variables except x as follows [26, 34].
For two continuous variables x and y, the marginal density function of y is obtained by integrating the joint distribution over variable x.
If the marginal distribution function of (Y_{i}) is denoted by (F^), we have {F^ ( y ) = frac{1}{alpha} int_{ - infty}^{y} {G ( u ),dF ( u )}, so the marginal density function of Y is {f^ ( y ) = frac{1}{alpha}G ( y )f ( y ).
Throughout, let (X n )n∈Nbe a sequence of absolutely continuous random variables on a fixed probability space Ω, F, P with the joint density function g1, n x1,..., x n ), n ∈ N, and f j (x), j = 1, 2,... be the the marginal density function of random variable X j. (k n )n∈Nbe a subsequence of positive integers, such that, for every ε > 0, ∑ n = 1 ∞ exp - k n ε < ∞.
Let (mathbf{X}=(X_{1},dots,X_{n})) and (mathbf{Y}=(Y_{1},dots,Y_{n})) be positive random vectors with respective marginal density functions (f_{i}) and (g_{i}), (i=1,dots,n).
The marginal density functions and the traceplots indicate that σ converges to a stationary level, i.e., the parameter values oscillate around a constant value so that we can assume that the parameter space has been explored exhaustively.
If we appeal to the copula theory for determining the joint PDF, the PDFs of the signals s(t) and n(t) should be considered as the marginal density functions in the copula discussion.
Let (mathbf{X}=(X_{1},dots,X_{n})) and (mathbf{Y}=(Y_{1},dots,Y_{n})) be random vectors with respective survival functions F̄ and Ḡ. Denote by (f_{i}) and (g_{i}), (i=1,dots,n), their marginal density functions, respectively.
Given two random variables, x and y, MI is computed by: 1 where p x, y), p(x), and p y) are joint density function and marginal density functions of x and y, respectively.
