My Favorite One-Liners: Part 100

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today’s quip is one that I’ll use surprisingly often:

If you ever meet a mathematician at a bar, ask him or her, “What is your favorite application of the Cauchy-Schwartz inequality?”

The point is that the Cauchy-Schwartz inequality arises surprisingly often in the undergraduate mathematics curriculum, and so I make a point to highlight it when I use it. For example, off the top of my head:

1. In trigonometry, the Cauchy-Schwartz inequality states that

|{\bf u} \cdot {\bf v}| \le \; \parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel

for all vectors {\bf u} and {\bf v}. Consequently,

-1 \le \displaystyle \frac{ {\bf u} \cdot {\bf v} } {\parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel} \le 1,

which means that the angle

\theta = \cos^{-1} \left( \displaystyle \frac{ {\bf u} \cdot {\bf v} } {\parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel} \right)

is defined. This is the measure of the angle between the two vectors {\bf u} and {\bf v}.

2. In probability and statistics, the standard deviation of a random variable X is defined as

\hbox{SD}(X) = \sqrt{E(X^2) - [E(X)]^2}.

The Cauchy-Schwartz inequality assures that the quantity under the square root is nonnegative, so that the standard deviation is actually defined. Also, the Cauchy-Schwartz inequality can be used to show that \hbox{SD}(X) = 0 implies that X is a constant almost surely.

3. Also in probability and statistics, the correlation between two random variables X and Y must satisfy

-1 \le \hbox{Corr}(X,Y) \le 1.

Furthermore, if \hbox{Corr}(X,Y)=1, then Y= aX +b for some constants a and b, where a > 0. On the other hand, if \hbox{Corr}(X,Y)=-1, if \hbox{Corr}(X,Y)=1, then Y= aX +b for some constants a and b, where a < 0.

Since I’m a mathematician, I guess my favorite application of the Cauchy-Schwartz inequality appears in my first professional article, where the inequality was used to confirm some new bounds that I derived with my graduate adviser.

Correlation and Causation: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on data sets that (hopefully) persuade students that correlation is not the same as causation.

Part 1: Piracy and global warming. Also, usage of Internet Explorer and murder.

Part 2: An xkcd comic.

Part 3: STEM spending and suicide. Consumption of margarine and divorce. Consumption of mozzarella and earning a doctorate. Marriage rates and deaths by drowning.

Part 4: Donna the Deer Lady.

 

 

 

 

Why Not to Trust Statistics

Math with Bad Drawings has an excellent post on how the blind use of descriptive statistics can be deceptive. I’ll definitely be sharing a few these with my students. Here’s one of several examples; I recommend reading the whole thing.

Correlation is not Causation (Part 4)

One of the standard topics in an undergraduate statistics course is the principle that two things that are highly correlated do not necessarily have a cause-and-effect relationship. Here is a hilarious example of this fallacy.

And, in case you’re wondering, here’s the rest of the story:

 

Five Ways to Lie with Charts

This is a cute article about ways that people can lie with charts: (1) Puzzling perspective, (2) Swindling shapes, (3) Trendsetters are tricksters (implying a false correlation), (4) Hiding in plain sight, and (5) Changing the scale of the axis.

http://nautil.us/issue/19/illusions/five-ways-to-lie-with-charts

Correlation and Causation: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on data sets that (hopefully) persuade students that correlation is not the same as causation.

Part 1: Piracy and global warming. Also, usage of Internet Explorer and murder.

Part 2: An xkcd comic.

Part 3: STEM spending and suicide. Consumption of margarine and divorce. Consumption of mozzarella and earning a doctorate. Marriage rates and deaths by drowning.

 

 

 

 

Correlation and causation

xkcdcorrelation

Source: http://www.xkcd.com/552/