Explanation of the formula for variance in probability theory.

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😎 Explain this formula:

Var(X) = Var( E(X|Y) ) + E( Var(X|Y) )
🤖 This formula is known as the Law of Total Variance, which helps break down the total variance of a random variable X into two components based on conditional expectations and variances.

– Var(X) represents the total variance of the random variable X.
– E(X|Y) represents the conditional expectation of X given Y, which is a function of Y and provides the expected value of X when we know the value of Y.
– Var(E(X|Y)) represents the variance of the conditional expectation E(X|Y), which measures how much E(X|Y) varies as Y changes.
– E(Var(X|Y)) represents the expected value of the conditional variance Var(X|Y), which measures how much the conditional variance of X changes as Y changes.

The Law of Total Variance states that the total variance of X can be decomposed into these two components: the variance of the conditional expectation plus the expected value of the conditional variance. It’s a fundamental result in probability and statistics used in various applications, such as understanding the total variability of a random variable in the presence of conditional information.

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