Probability distribution of the net present value, Advanced Statistics

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Suppose that $4 million is available for investment in three projects. The probability distribution of the net present value earned from each project depends on how much is invested in each project. Let I_t be the random variable denoting the net present value earned by project t. The distribution of I_t depends on the amount of money invested in project t, as shown in Table 2 (a zero investment in a project always earns a zero NPV). Use dynamic programming to determine an investment allocation that maximises the expected NPV obtained from the three investments.

Table

	Investment (millions)	Probability
Project 1	$1	P(I₁ = 2) = 0.6	P(I₁ = 4) = 0.3	P(I₁ = 5) = 0.1
	$2	P(I₁ = 4) = 0.5	P(I₁ = 6) = 0.3	P(I₁ = 8) = 0.2
	$3	P(I₁ = 6) = 0.4	P(I₁ = 7) = 0.5	P(I₁= 10) = 0.1
	$4	P(I₁ = 7) = 0.2	P(I₁ = 9) = 0.4	P(I₁= 10) = 0.4
Project 2	$1	P(I₂ = 1) = 0.5	P(I₂ = 2) = 0.4	P(I₂ = 4) = 0.1
	$2	P(I₂ = 3) = 0.4	P(I₂ = 5) = 0.4	P(I₂ = 6) = 0.2
	$3	P(I₂ = 4) = 0.3	P(I₂ = 6) = 0.3	P(I₂ = 8) = 0.4
	$4	P(I₂ = 3) = 0.4	P(I₂ = 8) = 0.3	P(I₂ = 9) = 0.3
Project 3	$1	P(I₃ = 0) = 0.2	P(I₃ = 4) = 0.6	P(I₃ = 5) = 0.2
	$2	P(I₃ = 4) = 0.4	P(I₃ = 6) = 0.4	P(I₃ = 7) = 0.2
	$3	P(I₃ = 5) = 0.3	P(I₃ = 7) = 0.4	P(I₃ = 8) = 0.3
	$4	P(I₃ = 6) = 0.1	P(I₃ = 8) = 0.5	P(I₃ = 9) = 0.4

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