How do you know if X and Y are independent random variables?
Intuitively, two random variables X and Y are independent if knowing the value of one of them does not change the probabilities for the other one. In other words, if X and Y are independent, we can write P(Y=y|X=x)=P(Y=y), for all x,y.
Are Gaussian random variables independent?
4 Answers. The answer is no. For example, if X is a standard random variable, then Y=−X follows the same statistics, but X and Y are clearly dependent. No, there is no reason to believe that any two standard gaussians are independent.
What is the PDF of a Gaussian random variable?
A complex random variable Z = X + jY is a pair of real random variables X and Y. The pdf of a complex RV is the joint pdf of its real and imaginary parts. If X and Y are jointly Gaussian, Z = X + jY is a complex Gaussian RV.
How are random variables x and Y independent?
Random variables X and Y are independent if their joint distribution function factors into the product of their marginal distribution functions • Theorem. Suppose X and Y are jointly continuous random variables. X and Y are independent if and only if given any two densities for X and Y their product is the joint density for the pair (X,Y) i.e.
How to find out if X and Y are independent?
My goal is to figure out if X and Y are independent. f X ( x) = ∫ 0 2 2 7 ( x + 2 y) d y = 2 7 ( x + 3). f Y ( y) = ∫ 0 1 2 7 ( x + 2 y) d x = 1 7 ( 4 y + 1) = 2 7 ( 2 y + 1 2). Does this mean X and Y are independent? So are we are expecting to find E ( X Y) ≠ E ( X) × E ( Y) ? No need to go to expectations.
What is the covariance of X and Y?
• For independent random variables X, Y E(XY) = E(X)E(Y) whenever these expectations exist. Proof: For random variables X, Y with E(X), E(Y) < ∞, the covariance of X and Y is • Covariance measures whether or not X-E(X) and Y-E(Y) have the same sign.