problem 3

(a) The new PDF is,

g (x) = 2 F (x) f (x)

$g (x) \geq 0$ in the same range as f

\begin{matrix} \int \infty - \inftyg (x) dx & = \int \infty - \infty 2 F (x) f (x) dx & (5.6) = \int \infty - \inftyd (F^{2} (x)) & (5.7) = lim x \to \infty F^{2} (x) - lim x \to - \infty F^{2} (x) & (5.8) = 1 - 0 & (5.9) \end{matrix}

(b) The new PDF is,

g (X) = \frac{1}{2} (f (x) + f (- x))

\begin{matrix} \int \infty - \inftyg (x) dx & = \frac{1}{2} \int \infty - \inftyf (x) dx + \frac{1}{2} \int \infty - \inftyf (x) dx & (5.10) = 1 & (5.11) \end{matrix}