How to calculate auto-correlation of a bpsk modulated signal or how to calculate expectation value of complex exponential function
This question is essentially unanswerable because too many details are left out.
Here is an answer to the following simplified version of the question


  What is the autocorrelation function of the random process 
  $$X(t) = A\exp(j2\pi f_ct), -\infty &lt; t &lt; \infty$$
  where $A$ is a random variable equally likely to have values
  $+1$ and $-1$, and the autocorrelation function is defined as
  $$R_X(t,\tau) = E[X(t+\tau/2)X^*(t-\tau/2)], -\infty &lt; t,\tau &lt; \infty.$$


This is easy since the only random variable involved is $A$, and
it is a discrete random variable.  Thus, we have
$$\begin{align}
E[X(t)]&amp;= E[A]\cdot \exp(j2\pi f_ct)\\
&amp;= 0 ~~\scriptstyle{\text{since }} E[A]=0.\\
R_X(t,\tau) &amp;= E[X(t+\tau/2)X^*(t-\tau/2)],\\
&amp;= E[A^2]\cdot \exp\left(j2\pi f_c(t+\tau/2)-j2\pi f_c(t-\tau/2)\right)\\
&amp;= \exp(j2\pi f_c\tau)~~\scriptstyle{\text{since }} E[A^2]=1.
\end{align}$$



What would be needed to make this question worth answering? First,
one usually has a baseband pulse shape, usually time-limited, 
that is nowhere mentioned by the OP. The autocorrelation function
of a BPSK signal is usually a scaled version of the
autocorrelation of the baseband pulse shape (and so the power
spectral density is determined by the shape of the baseband pulse).
To get to this result, one needs to consider not just
one bit but rather the stream of bits that modulate the BPSK signal,
and assume that these bits are i.i.d. random variables (such as
$A$ above). Second, something needs to be said about the noise
$n(t)$ that the OP casually includes in her question.