I know if I want to calculate autocorrelation of a random process , I have this rule :
$ R_X (t_1 , t_2) = E \{ X(t_1)X^*(t_2) \} $ .
In my cource I had this example :
$ X (t ) = A cos(2πft + Θ) $
A: constant. Θ: uniform in [0, 2π].
Find the autocorrelation of X.
in this case we but :
$ R_X (t_1 , t_2 ) = E [ A cos(2πft_1 + Θ) A cos(2πft_2 + Θ)] = A E [cos(2π (t_1 + t_2 ) + 2Θ) + cos(2πf (t_1 − t_2 ))] $
and he didn't say any thing about probability density function , so how he solved the example like this :
$= A cos(2πf (t1 − t 2 )) + A E [cos(2π (t1 + t 2 ) + 2Θ)]$
$E [cos(2π (t1 + t 2 ) + 2Θ)]=\frac{1}{2π}∫_{0} ^{2π}cos(2πf (t1 + t 2 ) + 2θ )dθ = 0.$
$RX (t_1 , t_2 ) = A cos(2πf (t_1 − t_2 ))$
so how can explain to my these questions :
1. why $ A E[ A cos(2πf (t_1 − t_2 )) ]=cos(2πf (t_1 − t_2 )) $ . 2. what I must conceder the PDF f_X(x) to solve $E [cos(2π (t1 + t 2 ) + 2Θ)]$ .