How to find the output mean and autocorrelation of a non-linear system I need help with this question. I am sure this is the right StackExchange forum for this type of question. 
Consider a nonlinear device such that the output is $Y(t) = aX^2(t)$, where the input X(t) consists of a signal plus a noise component, $X(t) = S(t) + N(t)$.
Determine the mean and autocorrelation function for $Y(t)$ when the signal $S(t)$ and the noise $N(t)$ are both Gaussian random processes and wide sense stationary (WSS) with zero mean, and $S(t)$ is independent of $N(t)$. 
I know that for a WSS signal, the mean and autocorrelation depends on the time difference, $\tau$ only. Meaning that the mean of $S(t)$ and $N(t)$ is a constant and their individual autocorrelation is not dependent on $t$. 
I've been working only with linear systems. I don't know how to solve this problem since $Y(t)$ a nonlinear system. I will be happy if you can help me with the solution 
UPDATE 1 
I followed the suggestion from @Hilmar and came up with the following: 
1. The mean of $Y(t) = 0$ since $S(t)$ and $N(t)$ have 0 mean. 
2. For the autocorrelation, I used this formula $R_{yy}(\tau)=R_{xx}(\tau)*h(\tau)*h(-\tau)$. 
looking for $R_{xx}$, I used $R_{xx}=E[X(t)X(t+\tau)]$. 
I ended up having $R_{xx} = E[S(t)S(t+\tau) +S(t)N(t+\tau)+N(t)S(t+\tau) + N(t)N(t+\tau)]$. Since $S(t)$ and $N(t)$ are independent, $R_{SN} = R_{NS} = 0$
$=&gt; R_{XX}= E[S(t)S(t+\tau) + N(t)N(t+\tau)]$ 
$=&gt; R_{XX} = R_{SS} + R_{NN}$

$R_{YY} = (R_{SS} + R_{NN})*h(\tau)*h(-\tau)$ 
Now I am stuck. From the definition of WSS signal, $R_{SS}$ and $R_{NN}$ are constants. i.e. they are dependent only on $\tau$. I can assume their sum to a some constant value, $C$I don't know how to move from here. I will need to find $h(\tau)$ I don't know how to go about it.
Put your second equations into your first equation, express $Y(t)$ as a function of $S(t)$ and $N(t)$
Apply the definition for mean and autocorrelation.
Simplify and solve