= Solution
Inverse <Fourier transform> of the <expected value> gives
$$
\boxed{\langle E(x,z)\rangle=e^{i\mu-\sigma^2/2}},
$$
independent of both coordinates. Its squared <modulus> is the coherent <scalar wave intensity> $e^{-\sigma^2}$, which must not be confused with the <mean wave intensity>.
For the latter use the stationary two-point statistics. Let $F_0(d\nu)$ be the uncentered <spectral measure of a stationary random field> of $E(0,z)$, normalized by
$$
\langle E(0,z+\zeta)E(0,z)^*\rangle=\int e^{i\nu\zeta}F_0(d\nu).
$$
Since $|E(0,z)|=1$, its total mass is $\int F_0(d\nu)=1$. Under free <paraxial propagation> each spectral component is multiplied by $e^{-i\nu^2x/(2k)}$; in its <wave-field correlation> this <wave phase> multiplies its <complex conjugate> and cancels. The <stationary intensity under free paraxial propagation> is consequently
$$
\boxed{\langle I(x,z)\rangle=\int F_0(d\nu)=1}.
$$
<Stationarity> is essential to this pointwise statement: it makes the spectral covariance diagonal. Conservation of integrated power alone would not show that a particular spatial point keeps the same <scalar wave intensity>.
The same result follows locally from $I_x+J_z=0$, where $J=\operatorname{Im}(E^*E_z)/k$. Taking <expectations> makes $\langle J\rangle$ independent of $z$, so $\partial_x\langle I\rangle=0$. Thus coherent <scalar wave intensity> and <mean wave intensity> are both constant in propagation, with different values. The incoherent contribution has mean $1-e^{-\sigma^2}$. Individual realizations can show focusing and changing <scalar wave intensity> even though the <expected value> remains constant. All these propagation statements concern the free <parabolic wave equation> from part (a).
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