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Which is a measure of the relationship of Two RVs?
1 2 dx = 1 3 E(XY)=E(X3)=0. Hence, cov(X,Y)=E(XY)− E(X)E(Y)=0, but Y is a function of X, so these two variables are not independent. Another measure of strength of relationship of two rvs is correlation.
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.
Is there a dependence on variance between X and Y?
X and Y are uncorrelated. On mean, Y doesn’t depend on X because whatever value is X, the mean of Y is 0. But clearly the distribution of values of Y depends on X value. In this case, for example, the variance of Y is 0 when X=0 and > 0 when X=1, thus there is, at least, a dependence on variance, i.e. there is a dependence.
When is the covariance of two independent variables zero?
Property 2 says that if two variables are independent, then their covariance is zero. This does not always work both ways, that is it does not mean that if the covariance is zero then the variables must be independent. The following small example shows this fact.
What is the theorem about variance and covariance?
Fact:If two RV’s are independent, we can’t predict one using the other, so there is no linear association, and their covariance is 0. Theorem:If X and Y are independent, then Var(X +Y) = Var(X)+Var(Y) In other words, if the random variables are independent, then the variance of their sum is the sum of their variances!
What does covariance mean when the mean is 0?
Note also that if one of the variables has mean 0, then the covariance is simply the expected product. Trivially, covariance is a symmetric operation. cov(X, Y) = cov(Y, X). As the name suggests, covariance generalizes variance.
How to calculate the covariance of X and Y?
The covariance of (X, Y) is defined by cov(X, Y) = E([X − E(X)][Y − E(Y)]) and, assuming the variances are positive, the correlation of (X, Y) is defined by cor(X, Y) = cov(X, Y) sd(X)sd(Y) If cov(X, Y) > 0 then X and Y are positively correlated. If cov(X, Y) < 0 then X and Y are negatively correlated. If…