Hello everybody

I have a bit of a problem with understanding the conversion from a WSCS process  X(t) to a WSS process  Y(t) = X(t - \Delta) . With  \Delta the time shift being a uniform random variable on  (0,T) , independent of  X(t) and  T being the period of the mean function of  X(t)

The problem begins with the method to find the mean function of  Y(t) :

 m_{Y} = E\{X(t - \Delta)\} = E\{E[X(t - \Delta)|\Delta]\} = E\{m_{X}(t - \Delta)\}

First, and it might seem very basic, I don't get the syntax  E[X(t - \Delta)|\Delta]

And second, why by averaging the mean of the WSCS process over its period  T would we get the mean function of the WSS process ?

If I understand that I could understand the same kind of process used to find the autocorrelation function of  Y(t)  from the autocorrelation function of  X(t)

Please help me !!