A new plant at a manufacturing site has to be first installed and then commissioned. The times required for these two stages depend upon different random factors, and can therefore be regarded as independent. Based on past experience, the respective distributions for X (installation time) and Y (commissioning time), both in days, are as follows:

Find the joint distribution for X and Y, and the probability that the total time will not exceed seven days.

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U 3 4 P(X=u) 0.1 0.4 5 0.3 6 0.2 Vj P(Y = v;) 2 0.50 3 0.35 4 0.15