The probability density function of a random variable \(x\) is given by

\[p(x)= \begin{cases}0 & \text { for } x<0 \\ 0.5 & \text { for } 0 \leq x \leq 2 \\ 0 & \text { for } x>2\end{cases}\]

Determine \(E[x], E\left[x^{2}\right]\), and \(\sigma_{x}\).