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4. GreEdy Algorithms. The rational knapsack problem can be framed as .pdf

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4. Given X1Geo(0.5),X2Geo(0.25),X3Geo(0.25),X4Geo(0.2),X5Geo(0.1) where xi=1,2,, (1) Find P(X1=4X1=2) ? (2) Find P(X14X12) ? (homework 3/20 revised) (3) Find P(X1>4X1>2) ? (homework 3/20 revised) (4) Find P(X33) ? (5) Find the Var(X5) of r.v. X5 (6) Find E(Y) and Y=X1+X2+X3+X4+X5.

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4. GreEdy Algorithms. The rational knapsack problem can be framed as a container that can hold a weight capacity C together with a list of provisions pi,i=0,,n to be placed in the container. Each provision has a corresponding weight wi and value vi. The provisions are to be placed into the container without exceeding the capacity C and such that the sum of values is maximized. (a) (5 pts) Describe a greedy heuristics that could be used to fill the container. (b) ( 5 pts) Assume that fractional amounts 0f1 of each provision can be used. Describe an optimal filling strategy and its time complexity. (c) (5 pts) Now assume an all or nothing (0 or 1) approach must be used. Describe an algorithm, and its complexity, that is guaranteed to find the maximum sum of values without exceeding the capacity C.

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