Why not any random variable is an unbiased estimator of the population mean?

To prove sample mean is an unbiased estimator of the population mean, we use the following step, $\mathbb{E}[\sum \frac{X_i}{n}] = \frac{1}{n} \sum \mathbb{E}[X_i] = \mu$, given that $\mathbb{E}[X_i] = \mu$.

Now my question is if we take say first random variable $X_1$ as an estimate we can go ahead and say that since $\mathbb{E}{[X_1]} = \mu$, $X_1$ is an unbiased estimator. Does that mean $X_1$ or for that matter any value is an unbiased estimator?

I know it is not. I am doing a conceptual mistake in thinking, please let me know where I am going completely wrong.

 As commented below for iid's every sample is indeed an unbiased estimate of the mean.

• The sample mean is not used to estimate the sample mean. That is a nonsense statement. If the X$_i$ are iid then any single observation is an unbiased estimator of the population mean (assuming it exists). The reason to take the average of all n is that it reduces the variance by a factor of 1/n. Nov 22, 2017 at 5:32
• Apologies, I meant population mean. Edited. Nov 22, 2017 at 5:35
• @MichaelChernick Also thanks, in case of iid's every random sample indeed is an estimation of sample mean!!! that is what I wanted to be assured of. Thanks!! Nov 22, 2017 at 5:36

Let a random variable $X$ have mean $\mu$. Then if $(x_1, \ldots x_n)$ are realizations from $X$, $\frac{1}{n} \sum x_i = \hat{\mu}$ is an unbiased estimator of $\mu$. This is because $\mathbb{E}[\mu - \hat{\mu}] = 0$, a fact that we will use later. In fact if the data is drawn IID from $X$ then each individual sample is an unbiased estimator of the mean; the reason we take many samples is to lower the variance of our estimate.
Now if, as you mentioned, we have multiple random variables, $X_1, \ldots, X_p$, each of which has mean $\mu$, then $\mathbb{E}\left[\frac{1}{p}\sum_{i=1}^p X_i \right] = \mu$ as you suggest. Furthermore, $\mathbb{E}X_1 = \mu$. That is, $\mathbb{E}[\mu - \mathbb{E}X_1]=0$. This implies that the observed realization of $X_1$ is an unbiased estimator of $\mu$. There is nothing startling here but it is distinct, I think, from the question you meant to be asking which was answered in the previous paragraph. This distinction is important and worth understanding well.
• Thanks! this was insightful, but I have a question, so if we consider the equation $\mathbb{E}[\mu - X_1]$, does not that imply zero too ? by linearity we can take $\mu - \mathbb{E}[X_1] = 0$. I understand that expectation of $X_1$ is also an unbiased estimator. Apologies if my questions are really naive. Nov 22, 2017 at 5:56
• Since $X_1$ is a random variable, therefore should not $\mu - X_1$, also be a random variable? Nov 22, 2017 at 6:16