What are orthogonal random variables?
Orthogonality is a property of two random variables that is useful for applications such as parameter estimation (Chapter 9) and signal estimation (Chapter 11). Definition: Orthogonal Random variables X and Y are orthogonal if .
How is LLSE calculated?
Linear Least Squares Estimate (LLSE) The LLSE of Y given X, denoted by L[Y|X], is the linear function a+bX that minimizes E(|Y −a−bX|2). C(g) = E(Y2 +a2 +b2X2 −2aY −2bY X +2abX) = E(Y2)+a2 +b2E(X2)−2aE(Y)−2bE(YX)+2abE(X).
When do two independent random variables become orthogonal?
And no, orthogonal does not mean that the two variables are independent. Further, independence does not imply orthogonality. But we can state the following: two independent random variables are orthogonal if, and only if, at least one of the two R.V. has mean: .
What does orthogonal RV mean for a random variable?
In case of random variables E[X] = ∫∞ − ∞xdμX so, orthogonal RVs are those with E[XY] = ∫∞ − ∞∫∞ − ∞xydμXdμY = 0 Orthogonal means the vectors are at perpendicular to each other. We state that by saying that vectors x and y are orthogonal if their dot product (aka inner product) is zero, i.e. x ⊺ y =0.
When does orthogonality and uncorrelatedness imply statistical independence?
Consequently, if at least one of the two RVs X and Y have a zero mean, then orthogonality implies uncorrelatedness and vice versa. Statistical independence means that the joint PDF of two random variables can be written as the product of the individual PDFs:
What does the expected value of orthogonality mean?
However for vectors with random components, the orthogonality condition is modified to be Expected Value E[x ⊺ y] = 0. This can be viewed as saying that for orthogonality, each random outcome of x ⊺ y may not be zero, sometimes positive, sometimes negative, possibly also zero, but Expected Value E[x ⊺ y] = 0.