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

Mathematics Asked on December 1, 2021

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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??? $langle x(t) rangle=A$

One Answer

I guess when you calculate the time average integral, you are suppose to treat r.v.'s as if they are constants:

$$bar{x} = lim limits_{T to infty} frac{1}{2T} int limits_{-T}^{T} x(t)~dt$$

$$bar{x} = lim limits_{T to infty} frac{1}{2T} int limits_{-T}^{T} A~dt$$

$$bar{x} = A ~lim limits_{T to infty} frac{1}{2T} int limits_{-T}^{T} 1~dt$$

$$bar{x} = A ~lim limits_{T to infty} frac{1}{2T} bigg[tbigg]_{-T}^{T} $$

$$bar{x} = A ~lim limits_{T to infty} frac{1}{2T} bigg[T--Tbigg]$$

$$bar{x} = A ~lim limits_{T to infty} frac{2T}{2T}$$

$$bar{x} = A ~lim limits_{T to infty} 1$$

$$bar{x} = A$$

Answered by pico on December 1, 2021

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