For a system that is characterized by the coordinate $\mathbf{r}$ and the conjugate momentum p, show that the expectation value of an operator $F$ can be expressed in terms of the Wigner distribution $W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right)$ as
$$
\langle F\rangle=\langle\Psi|F| \Psi\rangle=\iint F_W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right) W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right) d^3 r^{\prime} d^3 p^{\prime}
$$
where
$$
F_W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right)=\int e^{(d / /) \mathbf{p}^{\prime} \cdot \mathbf{r}^{\prime}}\left\langle\mathbf{r}^{\prime}-\frac{\mathbf{r}^{\prime \prime}}{2}|F| \mathbf{r}^{\prime}+\frac{\mathbf{r}^{\prime \prime}}{2}\right\rangle d^3 r^{\prime \prime}
$$
and where the function $W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right)$ is defined in Problem 5 in Chapter 3. Show that for the special cases $F=f(\mathbf{r})$ and $F=g(\mathbf{p})$ these formulas reduce to those obtained in Problems 5 and 6 in Chapter 3, that is, $F_W\left(\mathbf{r}^{\prime}\right)=f\left(\mathbf{r}^{\prime}\right)$ and $F_W\left(\mathbf{p}^{\prime}\right)=g\left(\mathbf{p}^{\prime}\right)$.