Question

Give a condition on two variables X and Y such that E[XY]=E[X]E[Y]. Suppose that X∼U({−1,0,1})

Answer 1

E[XY]=E[X]E[Y] when X and Y are independent random variables. For X~U({-1,0,1}), its independence from Y would mean that the joint probability factors into the product of their individual probabilities.

Step-by-step explanation:

The condition under which E[XY]=E[X]E[Y] holds is that the random variables X and Y must be independent. If X and Y are independent, then the expected value of their product is equal to the product of their expected values. The given distribution of X, which is X~U({-1,0,1}), represents a uniform distribution over the set {-1, 0, 1}. For independent random variables, the fundamental property that defines them is that the occurrence of one does not affect the probability of occurrence of the other, which allows us to factor the joint probability of X and Y into the product of their individual probabilities. Hence, P(XY) = P(X)P(Y) leads to E[XY] = E[X]E[Y]. To see if the given X and any Y are independent, one must check whether the joint distribution of X and Y factors into the product of their marginal distributions.

Here are examples of how the general principles of probability apply:

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• If Y and Z are independent, then P(Y AND Z) = P(Y)P(Z), which translates to independence influencing the combined probabilities.
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• The expected value, or mean (μ), of a discrete random variable can be calculated using the formula E(X) = μ = Σ xP(x), where you multiply each possible value of X by its probability and sum the products.