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I am confused at one point. The coin flipping Bernoulli Process has a probability of p of getting HEADS and a probability of 1-p of getting TAILS. Let's define a random variable x[n], which takes the value +1 when it is a HEADS, and -1 when it is a TAILS. The mean or estimation of x[n] becomes (2p -1) and I can derive it as an integration of probability function with the function itself. Which results in the sum of +1 times p and -1 time (1-p). However, when it comes to autocorrelation if the lag is 0, That is Estimate[x[n+m]x[n]] = 1 when m = 0 and if m is not equal to 0, it becomes (2p -1) ^ 2. I basically get it when I try to understand the physical meaning of it, but mathematically how this calculation is done? Because of the fact that the autocorrelation is a second-order property, it becomes (2p - 1) ^ 2. However, not for m = 0. Why and how?
E{x[n]} = 2p−1
E{x[n+m]x[n]} = 1 for [m = 0]
E{x[n+m]x[n]} = (2p - 1)^{2} for [m != 0]
E{x[n]} = 2p−1
E{x[n+m]x[n]} = 1 for [m = 0]
E{x[n+m]x[n]} = (2p - 1)^{2} for [m != 0]