consider a mix [0, 1] of a random payoff X and a fixed amount c. The payoff of a mix is X

consider a mix λ ∈ [0, 1] of a random payoff X and a fixed amount c. The payoff of a mix λ is Xλ = (1 − λ)X + λc. Let fλ denote the probability density function of Xλ. 

1. Consider a decision maker with u(z) = 2√ z, c = 0.42, X uniform on [0, 1]. What is the optimal λ?  

2. Consider a decision maker with u(z) = 2 log(1 + z), c = 0.42, X uniform on [0, 1]. What is the optimal λ? 

3. Consider a decision maker with u(z) = 2x 1+x , c = 0.42, X uniform on [0, 1]. What is the optimal λ?  

4. Consider a decision maker with u(z) = 2√ z, c = 0.42, X normal with mean 0.5 and variance 0.5. What is the optimal λ? 

5. Consider a decision maker with u(z) = 2 log(1 + z), c = 0.42, X normal with mean 0.5 and variance 0.5. What is the optimal λ?  

6. Consider a decision maker with u(z) = 2x 1+x , c = 0.42, X normal with mean 0.5 and variance 0.5. What is the optimal λ?  

7. What is the take-away lesson from this exercise?