= Solution
The deterministic drift vanishes at the positive stable point
$$
n_*=\sqrt{\frac{\lambda}{2\beta}},
\qquad
A'(n_*)=-4\beta n_*<0.
$$
When $n_*$ is large, fluctuations sample only a narrow range around it, so the <linear noise approximation> replaces
$$
A(n)\simeq A'(n_*)(n-n_*),
\qquad
B(n)\simeq B(n_*)=3\lambda.
$$
The resulting Ornstein-Uhlenbeck stationary density is the stated normal distribution with
$$
\boxed{\mu=n_*=\sqrt{\frac{\lambda}{2\beta}}},
$$
and
$$
\boxed{
\sigma^2=-\frac{B(n_*)}{2A'(n_*)}
=\frac{3\lambda}{8\beta n_*}
=\frac34\sqrt{\frac{\lambda}{2\beta}}
=\frac{3\mu}{4}
}.
$$
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