Final answer:
The generalized variance formula for the linear combination of two random variables X and Y, var[aX+bY], is proven by expanding the expected square deviation of aX+bY from its mean and simplifying using the definitions of variance and covariance.
Step-by-step explanation:
To prove the generalization of the variance formula var[aX+bY]=a²var[X]+b²var[Y]+2abcov[X,Y], we need to apply the definitions of variance and covariance. The variance of a random variable is the average of the squared deviations from the mean, and the covariance between two random variables measures how much the variables change together.
For two random variables X and Y, where a and b are constants, we can express the variance of their linear combination as follows:
- var[aX+bY] = E[((aX+bY) - E[aX+bY])^2]
- var[aX+bY] = E[(aX + bY - aE[X] - bE[Y])^2]
- var[aX+bY] = E[(a(X - E[X]) + b(Y - E[Y]))^2]
- var[aX+bY] = E[a^2(X - E[X])^2 + 2ab(X - E[X])(Y - E[Y]) + b^2(Y - E[Y])^2]
- var[aX+bY] = a^2E[(X - E[X])^2] + 2abE[(X - E[X])(Y - E[Y])] + b^2E[(Y - E[Y])^2]
- var[aX+bY] = a^2var[X] + 2ab·cov[X,Y] + b^2var[Y]
The final step follows from the facts that E[(X - E[X])^2] is the variance of X, E[(Y - E[Y])^2] is the variance of Y, and E[(X - E[X])(Y - E[Y])] is the covariance between X and Y.