(Solved): A candy mix includes three different types of candy. Let \( X \) and \( Y \) be the proportions of ...

A candy mix includes three different types of candy. Let \( X \) and \( Y \) be the proportions of the first two types of candy in the mixture. Suppose that the joint pdf of \( X \) and \( Y \) is uniform over the region where \( 0 \leq x \leq 1,0 \leq y \leq 1 \), and \( 0 \leq x+y \leq 1 \), and is 0 , elsewhere. (a) Find the pdf of \( W=X+Y \), the proportion of the first two types of candy in the mixture. (b) How likely is it that \( W \) will be greater than \( 0.5 \) ?