Time Average Mean of X(t)=A, where A is a r.v. Ergodic vs. non-ergodic.

Question

Time average of a sample function is defined as:
$$bar{x} = langle~x(t)~rangle = lim limits_{T to infty}frac{1}{2T} int limits_{-T}^{T} x(t) ~ dt$$
This is how I see it:  A few sample functions of X(t)=A, would be:
$$x_1(t) =0.2$$
$$x_2(t) = 0.7$$
$$cdots$$
$$text{etc}$$
How do they come up with 'A' as the time average???  $langle x(t) rangle=A$

pico · Answer

I guess when you calculate the time average integral, you are suppose to treat r.v.'s as if they are constants:
$$bar{x} = lim limits_{T to infty} frac{1}{2T} int limits_{-T}^{T} x(t)~dt$$
$$bar{x} = lim limits_{T to infty} frac{1}{2T} int limits_{-T}^{T} A~dt$$
$$bar{x} = A ~lim limits_{T to infty} frac{1}{2T} int limits_{-T}^{T} 1~dt$$
$$bar{x} = A ~lim limits_{T to infty} frac{1}{2T} bigg[tbigg]_{-T}^{T} $$
$$bar{x} = A ~lim limits_{T to infty} frac{1}{2T} bigg[T--Tbigg]$$
$$bar{x} = A ~lim limits_{T to infty} frac{2T}{2T}$$
$$bar{x} = A ~lim limits_{T to infty} 1$$
$$bar{x} = A$$

Time Average Mean of X(t)=A, where A is a r.v. Ergodic vs. non-ergodic.

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