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Given a time series equation:$$x_t=\beta_1+\beta_2t+w_t$$ where $w_t$ is white noise.

When I am calulateing the mean:$$E[x_t]=E[\beta_1+\beta_2t+w_t]$$ $$=\beta_1+\beta_2t+E[w_t]$$ $$=\beta_1+\beta_2t$$

However, isn't the varibale $t$, also a random variable s.t. the answer should be $$E[x_t]=\beta_1+\beta_2E[t]$$?

Edit: the whole question is:

Consider the time series $x_t=\beta_1+\beta_2t+w_t$, where $\beta_1$ and $\beta_2$ are known constants and $w_t$ is a white noise process with variance $\sigma^2$.

GarlicSTAT
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    Is $t$ *random*, or is it a deterministic sequence? – Alexis Feb 11 '20 at 15:12
  • @Alexis edited. – GarlicSTAT Feb 11 '20 at 15:17
  • @Tim that's interesting – GarlicSTAT Feb 11 '20 at 15:32
  • @Tim It's hard to see how modeling the time in a time series as a random variable would accomplish anything. Given that when one calls data a "time series" it means [there is an equal interval between successive times,](https://stats.stackexchange.com/a/126830/919) the only freedom left to introduce randomness lies in the start time and the common interval. I suspect extremely few people have ever seen an application where it was useful to model the start or the interval as a random variable. – whuber Feb 11 '20 at 16:01
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    @whuber I missed it's time... – Tim Feb 11 '20 at 16:22

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