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To solve the problem numerically, we need to work through the following steps: ### Joint Probability Distribution Table: | X/Y | 1 | 2 | 3 | |------|-----|-----|-----| | 1 | 0.2 | 0.2 | 0.1 | | 2 | 0.1 | 0.3 | 0.1 | We are given the joint probability distribution of \(X\) and \(Y\). ### Part (a) The Marginal Distributions #### Marginal Distribution of X: The marginal distribution of \(X\) is obtained by summing the joint probabilities for each value of \(X\) across all values of \(Y\). \[ P(X = 1) = P(X = 1, Y = 1) + P(X = 1, Y = 2) + P(X = 1, Y = 3) \] \[ P(X = 1) = 0.2 + 0.2 + 0.1 = 0.5 \] \[ P(X = 2) = P(X = 2, Y = 1) + P(X = 2, Y = 2) + P(X = ______ ______ ________ _________ _________ ________ __________ _____ ___ ____ _____ __________.
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