Why do we use time average and not ensemble average?
An ensemble average is a convenient theoretical concept since it is directly related to the probability density functions, which can be generally obtained by the theoretical analysis of a given physical system. On the other hand, a time average is more directly related to real experiments.
What is ensemble average in random process?
The ensemble is defined as a set of all possible outcomes of a stochastic process, and ensemble average means the expected object (like expected value for random variable) of the stochastic process. Simply speaking it is just the expected value of random variable, but defined for a more general abstract setup.