For each of the densities below, indicate whether the mean and variance of the associated random variable exist. In addition, find the median and mode, and indicate whether or not each density is symmetric.

(a) \(f(x)=3 x^{2} I_{[0,1]}|x|\)

(b) \(f(x)=2 x^{-3} I_{[1, \infty)}(x)\)

(c) \(f(x)=\left[\pi\left(1+x^{2}ight)ight]^{-1} I_{(-\infty, \infty)}(x)\)

(d) \(f(x)=\left(\begin{array}{l}4 \\ x\end{array}ight)(.2)^{x}(.8)^{4-x} I_{(0,1,2,3,4)}(x)\)