Season 7 Episode 6

Variance 

The variance of a random variable $X$ is defined as $\mathbb{E}(X^2)-\mathbb{E}(X)^2$. The $\mathbb{E}$ in there means that we calculate the expected or average value of an expression. When you first see it, this looks like it might be zero. But it's (usually) not zero! We can prove that it's not negative (it would be awkward if it were negative, because people usually call it $\sigma^2$, and call the square root the standard deviation!).

Consider a graph of $x^2$ with points for our data, weighted by how frequently each value occurs in our data. The center of mass for those points must be on or above the curve, because the graph of $x^2$ curves upwards. The $x$-coordinate of the center of mass is $\mathbb{E}(X)$ and the $y$-coordinate is $\mathbb{E}(X^2)$. The face that the center of mass lies above or on the curve means that $\mathbb{E}(X^2)\geq\mathbb{E}(X)^2$, so $\mathbb{E}(X^2)-\mathbb{E}(X)^2\geq 0$.

The argument above is a special case of Jensen’s inequality, which you can learn about at this webpage. I like to say that Jensen's inequality is a sort of inequality factory; you give it a function and it generates an inequality for you!

That link goes to one page in an open access book on probability, statistics, and random processes. The book is aimed at undergraduate students, but for some people in school who are reading this, that might be the take on probability that you’re looking for.

If you’d like to learn or revise variance (for A-level or equivalent), see for example this revision website.

Graphs and Networks

You can read more about graphs and networks on Mathigon. Mathigon started out as the textbook of the future, with courses on many different topics, but now it's so much more; you might also like Polypad or this interactive Timeline of Mathematics.

Friendship Paradox 

You can read more on Wikipedia, of course, or you can read the original paper on this topic here.

It's a paradox because the answer doesn't line up with people's expectations. Here's a paper about those expectations; What Makes You Think You're so Popular? Self-Evaluation Maintenance and the Subjective Side of the "Friendship Paradox".

And here are just two of many papers on the applications of the friendship paradox;

• Using the friendship paradox to sample a social network
• Social Network Sensors for Early Detection of Contagious Outbreaks

Friends of Friends of Friends... 

On the stream we looked at the average number of friends of friends. The obvious extension is to think about friends of friends of friends (one degree further!). You might like to try to work out an expression, in terms of expectations, for what that means.

If you keep taking the average again and again, then something interesting happens in the limit. You might like to investigate, perhaps by using a computer to calculate the average number of friends of friends of friends each person had in our example friendship network, or one of your own invention!