Intuitive understanding of covariance Note :  I have just started learning statistics and the question might be very basic. I know what is variance, standard deviation.
I have following 2 questions regarding covariance and correlation

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*What is the intuitive meaning of the product in the formula of Cov(X,Y). Why not add the two terms instead of multiplying ?


*In addition to direction, why can't I use the magnitude of Cov(X,Y) to determine the strength of relation ? I know this is because of random variables being in different scales but I still lack the intuition to why this is problematic ?
Note : I understand what correlation and covariance means but I don't understand the intuition behind the formula, meaning, why this particular formula gives the meaning that these terms have.

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*Suppose we define covariance by adding the two terms instead of multiplying them. Assume X and Y follow each other very closely (also, for simplicity, assume we normalize the values by centring them so that both have a mean of zero). Following each other means when X is positive, Y is also very likely to be positive. The same holds when X is negative. In this case, you might end up with a correlation of X and Y being very close to zero, while, as we assumed, these two variables move very closely with each other. The reason is that when X and Y are both positive, their summation is also positive, and when they are negative, their summation is also negative, so when we take the expectation of the summations, these positive and negative values might cancel out each other, resulting in a correlation value of very close to zero! That's why we use the product operator rather than summation to define covariance.


*The magnitude of the covariance is a function of the correspondence between two variables as well as the scale of two variables. So if you see a large covariance, you cannot say whether this is a sign of a high degree of correspondence between two variables or just the artefact of working with two variables with a large scale.
