lambdaphagy:

phi-of-two:

lambdaphagy:

rudescience:

The Average of Two Square Roots is Less than or Equal to the Square Root of the Avarage

File this under simple math things that make me unreasonably angry but are absolutely fascinating.

I mean… look at that… why???

Jensen’s inequality.

The QA mean is obvious too. It’s also easily solvable if you *haven’t* memorised a bunch of inequalities - just square both sides, multiply by four, and collect the terms on the right. You now have ( sqrt(x) - sqrt(y) )^2 >= 0 which is obviously true for all nonnegative x and y.

Sure.  The point isn’t so much to memorize a bunch of theorems, though, as to understand why they’re true and deploy them as needed.  Jensen’s inequality has a nice, almost trivial interpretation in terms of probability, so it’s hard to forget it (imho).  

Looking at it this way allows you to see why it’s true before you imperil yourself with algebra.  It also places the statement in a family of related results.  For example, does the result still hold with cube roots?  With seventeen variables?  Jensen tells you immediately that the answer must be yes.

Yeah, I think the importance of knowing Jensen is the understanding that this kind of fact results from the concavity of the square root function, and you’d get the opposite with a convex function.  It’s fundamentally a fact about concavity and convexity, and this can motivate choices in more general settings (”I don’t know what this function is specifically, but as long as I can show it has to be convex … ”)

(via lambdaphagy)