Two methods of measuring surface smoothness are used to evaluate a paper product. The measurements are recorded as deviations from the nominal surface smoothness in coded units. The joint probability distribution of the two measurements is a uniform distribution over the region 0 < ? < 4, 0 < ?, and ? − 1 < ? < ? + 1. That is, f (x, y) = c for x and y in the region. (a) Determine the value for c such that f (x, y) is a joint probability density function. (b) Marginal probability distribution of X (c) Conditional probability distribution of Y given X = 1 (d) ?(? < 5|? = 1)

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