You have a collection of investments with varying degrees of cash flow and uncertainty/risk. The cash flows (revenues) of these investments can be modeled as Xk ~N(k, ) for k = 1, ..., n. Mixing of investments is a portfolio optimization strategy that can be used to mitigate/dilute risk (anal- ogous to mixing liquid streams with different concentrations to achieve a target concentration). Specifically, you want to determine the mix proportions wk [0, 1], k = 1, ..., n (with 1 Wk 1) such that the total cash flow = n Y = Wk Xk . k=1 has an acceptable level of risk. Here, risk is defined as the probability that the total cash flow is less than a given threshold. (a) Consider n =3 investments with the following statistics: Investment 1 ( = 5 105 USD, = 1 105 USD) Investment 2 (2 = 4 105 USD, 2 = 1 10 USD) Investment 3 (3 = 1 106 USD, 03 5 105 USD). = Determine the risk of the individual investments (mixtures with proportions (1,0,0), (0, 1, 0), (0, 0, 1)). The threshold value for your risk is 4 105 USD. Provide some intuition as to why some investments lead to higher/lower risk than others. (b) Play around with mixtures of different proportions and determine whether mixing portfolios can indeed help reduce risk of the individual investments.