Expectation vs average value

[nigarabbasova]

Hi Anders!

This might be a bit of a silly question to ask, but I would like to double-check it nonetheless. Is it okay to use the terms expectation value and average value interchangeably? I have been a bit sloppy with it throughout the entire report, and would like to know what you think of it (and how you distinguish those two) when it comes to computing average/expectation values analytically vs numerically. I remember how you compared the two during a lecture on Project 4, but I am a bit unsure of how to apply that in the report.

Thank you! :)

[anderkve]

Hi @nigarabbasova!

Not at all a silly question -- it's a very good one! I'll reply after I'm done with some meetings.

[anderkve]

Hi again, @nigarabbasova!

First, let me say that it's not a big deal for this project if you mix the terms expectation value and average, so don't worry too much about it. But to as correct as possible, I would distinguish the terms in the following way:

• Expectation value: This is the "true", or let's say "theoretical", expected $x$ value, denoted $E(x)$ or $\langle x \rangle$, given some probability distribution $p(x)$. It can also be referred to as the mean of the pdf $p(x)$. The key point is that the expectation value is a property of the pdf, whether or not we know its exact value. So its value does not depend on questions like how many $x$ samples we draw from $p(x)$, etc. In problem 1, when you derive analytical expressions for e.g. $\langle \epsilon \rangle$ and $\langle \epsilon^2 \rangle$, it's therefore most correct to call these expectation values or means.

• Average: There are many different types of averages. (E.g. in a lecture I briefly discussed the definition from calculus for a function average, which isn't too relevant for Project 4.) The type of average that is relevant for this project is of course sample average or sample mean, that is, the good ol' $\bar{x} = \frac{1}{N} \sum xi$. In contrast to an expectation value, the sample average is a property of a dataset. The proper connection between the two is that the sample average is an estimator for the (typically unknown) expectation value. (This is why the project description talks about e.g. "computing numerical estimates of $\langle \epsilon \rangle$" rather than just "computing $\langle \epsilon \rangle$ numerically".) The two quantities are only the same in the limit of infinite samples, i.e. $\lim{N \rightarrow \infty} \bar{x} = \langle x \rangle$. So e.g. in Problem 5 a), it would technically be more correct to use a notation like $\bar{\epsilon}$ rather than $\langle \epsilon \rangle$ on the y axis of your plot. But we certainly won't be picky about that. :)

[nigarabbasova]

Thank you so much for such a detailed explanation - for someone with minimal theoretical background knowledge in statistics this was of great help! :)