# Notations question regarding the equation for sample mean and variance

Let $$X$$ be the outcome of a chance experiment with $$E(X) = \mu$$ and $$V(X) = \sigma^2$$. When $$\mu$$ and $$\sigma^2$$ can be estimated by repeating the experiment $$n$$ times with outcomes $$x_1, x_2,...,x_n$$, sample mean and variance are given by: $$\bar x = \frac{1}{n}\sum_{i=1}^{n}x_i$$ $$s^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i-\bar x)^2$$

This is followed by problems asking to prove $$E(\bar x)=\mu$$, $$E((\bar x-\mu)^2)=\sigma^2/n$$, $$E(s^s)=\frac{n-1}{n}\sigma^2$$. My question is notation related, should I be looking at $$x_1, x_2,...x_n$$ and $$\bar x$$ as random variables (which are typically denoted by capital letters in my book) each with $$\mu$$, $$\sigma^2$$, not actual outcomes of a specific sample (lower case letters are typically used to represent values i.e. $$P(X = x)$$)? This is a bit ambiguous since from the equations for $$\bar x$$ and $$s^2$$ I think (?) it could be interpreted both ways but in the proof it doesn't make sense if we are looking at a specific sample.

• May be relevant: math.stackexchange.com/questions/435846/… – NoChance May 21 at 0:33
• I think you have to do exactly what you deduced in your question. For instance, it is trivial to see that if you do this the fact that $E(\bar{x})=\mu$ becomes immediate. – Ariel Serranoni May 21 at 0:34

Yes, the $$x_i$$ are independent random variables having the distribution of $$X$$. Thus $$\overline{x}$$, which is defined in terms of the $$x_i$$’s, is a random variable, and so is $$s^2$$.