(Solved): Consider the following probability density function of a random variable \( X \) : \[ f(x)=\left\{ ...

Consider the following probability density function of a random variable \( X \) : \[ f(x)=\left\{\begin{array}{ll} 0.25 & \text { if } x=2 \text { or } 4 \\ 0.5 & \text { if } x=3 \\ 0 & \text { otherwise } \end{array}\right. \] (a) Find \( E u(X) \quad \) given \( u(x)=\ln x \). (b) Find \( E u(X) \) given \( u(x)=x^{2} \). (c) Provide an example of another random variable \( (Y) \) that is a meanpreserving contraction of \( X \). (d) Find \( \underline{\underline{E u}}(Y) \) given \( u(x)=\ln x \). (e) Find \( \underline{\underline{E u}}(Y) \) given \( u(x)=x^{2} \).