General question: How do you intuit $\Bbb P(|X - \Bbb E X| > a) \leq \text{Var}(X)/a^2$. Obviously the RHS and LHS are related, variance of a random variable is a measure of how much it deviates from it's expectation. Why is the inequality obvious?
@Thorgott You're just going to rewrite me a proof of Markov, which defeats the purpose of the question
(1) Markov is a calculation lemma, it is useless to interpret it (2) There is a significant and nonobvious gap in going from Markov to Chebyshev, Chernoff, etc... which is to modify the event $X > a$ to an event $f(X) > f(a)$ for some appropriate $f$
(2) is the magic trick
a calculation and a magic trick does not explain the final result
It's not useless to interpret. I'm saying that this is so close to tautologous that the actual proof and the intuition are basically the same. If the probability that $X\ge a$ is $c$, then your expected value will be at least $ca$. That's basically just how expectation works. It's a quantitative description of the idea that if you know where a distribution is centered, it can't grow much larger too often.
I agree (1) and (2) are both calculation. (2) is magic trick because it applies to squares and exponentials both, and somehow gives tighter bounds if you take $e^{\lambda X}$, and then optimize over $\lambda$, like Chernoff does
(2) is in this sense more of a mnemonic than (1), which is just pure calculation. (1) is just how integration works
It is easy to write down a proof as an answer to "why is blah inequality intuitively clear?". For example, suppose $X_1, \cdots, X_n$ are iid random variables, and $Y_1, \cdots, Y_n$ are iid copies of these iid random variables. Then $\text{Var} f(X_1, \cdots, X_n) \leq \sum_{i = 1}^n \Bbb E[f(X_1, \cdots, X_i, \cdots, X_n) - f(X_1, \cdots, Y_i, \cdots, Y_n)]_+^2$
I don't know why you think proof and intuition are mutually exclusive. The Markov inequality is so simple to prove that I don't know if I can say anything about it without that being equivalent to a proof once converted into formal language.
the fluctuations of $f(X_1, \cdots, X_n)$ from it's central tendency is measured by how much it depends on each of the variables $X_i$ --- but what is the inequality until you write it down?
I don't think Chernoff is surprising. You're fabricating a family of bounds and pick the best one. That ought to give you a better bound than just taking one. It's a clever and non-obvious trick for sure, but it's also just calculation.
@Thorgott But there's a point, which because clearer if you just forget the proof and the $e^X$ trick.
Having exponential moment gives you exponentially decaying tails.
Any pure statistician can tell you this without knowledge of Chernoff
People didn't just cook up the trick out of thin air. They knew it should happen, so they decided it's better to compare $e^{\lambda X}$ with $e^{\lambda a}$ than $X$ and $a$
This happens all the time in probability. Some inequality should happen, so you just have to find a way to get there by some obvious inequalities (like Markov)
What does "having exponential moment" mean? You make it sound like an intrinsic thing, but it's a trick where we replace the moment with the moment of sth else
Having exponential moment means the random variable has the Laplace transform, $\Bbb E\exp(\lambda X)$ exists. Or we think about it as having the moment generating function
"A site is a presentation of a sheaf topos as a structure freely generated under colimits from a category, subject to the relation that certain covering colimits are preserved." here you go
his standard response when people tells him things are wrong is, in thick russian accent: SO WHAT? INTERESTING EXAMPLE JAJAJA
according to people i know who know gromov
i think thats what he said when Yau disproved some conjecture of his
theres also the story of how Gromov once said there are only 5 examples of some thing (this is probably in PDR) and curt mcmullen was trying hard to find the 5th one until he found infinitely many
Both are of maximal rank, both have a similar local form by the rank theorem, both are one set-theoretic and one topological property apart from being as nice as you could want them to be
I'm not even fully one chapter into this book, I just want to rub my two brain cells together and go "hee hee it do easy thing with fiber but immersion bad"
and then if we have light with a frequency distribution $f(x)$ where $x$ is the wavelength, we can have another function $g$ which has the same response to all three cones
(now forget about the physics)
i.e. $\int_\mathbb{R} c_1(x)f(x)dx=\int_\mathbb{R} c_1(x)g(x)dx$ and the same for $c_2$ and $c_3$
intuitively, I believe that there are many of these functions. But what if we require them to be "nice", which sense of nice defines an interesting equivalence relation?
Furthermore, for some measure of size in the set of functions from R to R, what is the size of the set of functions with the same color as $f$?
