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What is the best probability book?
Must cover the basics:
- finite sample spaces
- conditional probability and independence
- unidimensional random variables
- random variables functions
- random variables of two or more dimensions
- discrete random variables (Poisson, Pascal, Multinomial, etc.)
- continuous random variables
You know, the usual. The course has an emphasis on applications for engineering, it seems. But anything is fine, as long as it really teaches probability.
The Apostol/Spivak of probability is what i'm asking, more or less.
Any scientists around?
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So, oil is created by animal remains that are subjected to sedimentary processes, and then brought to a simmer by metamorphic processes, correct -- or close enough?
How exactly does this fit into modern science? How was the arctic so livable millions of years ago?
Is evolution wrong? Is global warming wrong? Is Earth crust displacement true?
Or is the arctic just a massive sea animal graveyard where everything goes to die? Even so, wouldn't the crustaceans just eat everything eventually? The bones would simply be eroded into sediments before they could be subjected to such heat and pressure as to form oil.
Unless the idea is that they melt out of the sediments and pool, but, if that were the case, then why wouldn't the liquified bones, etc, simply continue heating until they were broken down entirely and their molecules scattered?
How does oil in the arctic make any sense?
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If scientists ever found a way of manipulating chromosomes, could this lead to a possible cure for down's syndrome, autism, and aspergers?
Hell, could this lead to humans finally having the ability to modify themselves to how they desire by changing their race, gender, and possibly even their sexuality?
Heine Borel and nutty intervals
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Yo /sci/ so whilst learning about topology I got up to the section on the general definition of compactness. For a bit of context I thought I'd have a look at Hardy's proof of Heine-Borel. One of the examples Hardy gives us is the set of all closed balls centre p/q rational in [0,1], radius ?/q^3. Obviously for some values of ? this is a closed cover of [0,1].
Then he tells us to consider the maximum length of a finite subcover: Turns out this is 2?*(1+1/2^3+2/3^3+3/4^3+...).
Obviously the series converges to a finite value, so for sufficiently small ? this is less than one. Thus we can't find a finite subcover so we can pick a ?>0 such that the set of intervals doesn't cover [0,1].
I must say, I guess it's sinking in but feels highly counterintuitive - you'd think you could, given any r in [0,1] find an interval of that form for any ?.
So my question is, /sci/, what is the maximum possible value of ? for which we won't have a cover? And which values will be missed?
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I didn't have an education growing up in extreme destitution. I didn't even attend school until I was 14. I'm in university now (passing math literacy tests and getting my HS diploma) at 23.
Everything I've been learning thus far has been a smooth entry into my brain. Everything is going great, but I struggle with the rudimentary maths you must have learned in late childhood.
Basically I can do calculus, matrix manipulation, logic, etc but I struggle with stuff like Pythagorean theorem, factoring, combinatorics, etc.
I just finished my first matrix test today and it went smooth except for one problem. I had to solve for a product of three matrices and the final question was "solve for k in the product", the product being 4k^3 + 12k^2 + 4 = 0.
I flubbed the question because I didn't know how to factor things out and solve for k.
This may sound stupid as shit, but can /sci/ suggest some elementary math texts I can quickly catch up in my weak spots?