autobot.d said:Is there a reference you could point me to or a reason why?
Would it be true if
[itex] \mu > \hat{\mu} [/itex]
and for
[itex] \mu < \hat{\mu} [/itex]
[itex] E[|\mu - \hat{\mu}|] = \frac{2}{\sqrt{\pi}} [/itex]
by going through and doing the actual integration.
The symbol $\mu$ represents the true population mean, while $\hat{\mu}$ represents the sample mean. When evaluating expectations, we are trying to estimate the true population mean using our sample mean.
To calculate the expected value for $\mu$ and $\hat{\mu}$, we take the average of all possible values that they can take on. For example, for $\mu$, we would take the average of all possible values in the population, while for $\hat{\mu}$, we would take the average of all possible values in the sample.
The sample size plays a crucial role in evaluating expectations for $\mu$ and $\hat{\mu}$. As the sample size increases, the sample mean ($\hat{\mu}$) becomes a better estimate of the true population mean ($\mu$). This is because with a larger sample size, we have more data points and a better representation of the population.
Yes, it is possible for the expected value of $\hat{\mu}$ to be equal to the true population mean $\mu$. This is more likely to happen when we have a large sample size, as the sample mean becomes a more accurate estimate of the true population mean.
The expected value of $\hat{\mu}$ gives us an idea of how close our sample mean is to the true population mean on average. It serves as an estimate of the true population mean, and the closer the expected value is to $\mu$, the better our estimate is likely to be.