Let {X(t), t ∈ R} be a continuous-time random process. The time average mean of X(t) is defined as (assuming that the limit exists in mean-square sense)

Consider the random process {X(t), t ∈ R} defined as

where U ∼ Uniform(0, 2π). Find ⟨X(t)⟩.

(X(t)) = lim T [ 1 2T T -T X(t)dt