Consider the continuous probability distribution & the reward function provided by the equations below:

fX (x) =

{x 1 2 x 1 x 1 1 x 2 0 else }

R(x) = { x 0 x x ^2 x 0 }

(a) Draw the probability density function (b) Draw the reward function (c) Calculate the expected reward (d) Assuming this was a betting game, would you take the bet based on the expected reward? (e) Assume that the probability distribution changed to the one below, would you still bet (assume reward function remains the same)?

fX (x) =

{ x 1 1 x 0 x 1 1 x 2 0 else }

Please write step by step and explain everything. Thanks.