(Solution) - Y takes on the value 2 with probability 1 Show -(2025 Original AI-Free Solution)

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Y takes on the value 2 with probability 1. Show that this is the case with a = 0 as in Theorem 7.3. What happens when you try to compute the correlation? Explain why (think of correlation as information).
Suppose the random variable X takes on the values 0, 1, and 2 with probabilities 0.2, 0.3, and 0.5, respectively. For each of the following random variables Y, find the joint distribution of X and Y, compute the covariance and the correlation, and sketch a graph showing the relationship, between X and Y that includes the range of values X = 0, X = 1, and X = 2.