Answer

White noise $n(t)$ PSD $N_0/2$ flat, $R(\tau)=N_0/2\delta(\tau)$.

In AM, received $y=A_c[1+\mu m(t)]\cos\omega_c t + n(t)$. After envelope detection, noise in-phase $n_I$ and quadrature $n_Q$.

Input SNR $\gamma_i=(A_c^2/2)/(N_0B)$, $B=2f_m$.

Output SNR $\gamma_o = \frac{\mu^2P_m}{1+\mu^2P_m}\gamma_i$, figure $\gamma<1$ (AM wastes carrier). DSB-SC $\gamma=1$.

Derivation: noise power $N_0B$, signal $A_c^2/2+\overline{m^2}/2$, LPF extracts baseband. Threshold ~10dB.