User:Simsilikesims/Probability theory

The general theory of probability is stated easily, so that any fool idiot could understand it. Let $\Omega$ be an abstract topological hypocaustic infiniumial supreminalial ontological vector space admitting a Borelisable Lindfield-bounded, continuous antinondifferentiable superluminal trachyodermic space, and let the space ${\mathcal {A}}$ be the space such that $A\in {\mathcal {A}}$ implies $A\subseteq \Omega$ and which admits a homomorphic field onto the real line. We can then proceed to introduce the essential structure of probabilty theory.

Elementary Definitions

Definition: A probabilizable space $(\Omega ,{\mathcal {A}})$ is a space such that the morphogenetic field induced on the space exists and is almost everywhere positive.

Definition: A probability measuration $\mu$ is a nonnegative nonunreal-valued measuration such that, given a premeasuration $\mu '$ there exists a sequence $A_{1},\dots ,A_{n},\dots$ with each $A_{n}\in {\mathcal {A}}$ and such that the sequence $\mu '(A_{n})$ with $A_{n}\subset A\subseteq \Omega$ and $A_{n-1}\subseteq A_{n}$ and a sequence $B_{1},\dots ,B_{n},\dots$ such that $B_{n}\in {\mathcal {A}}$ where if $A\subseteq B_{n}$ and $B_{n}\subseteq B_{n-1}$ there exists ${\text{sup lim}}_{n\to \infty }\mu '(A_{n})$ and ${\text{inf lim}}_{n\to \infty }\mu '(B_{n})$ and if these limits are equal for all $A\in {\mathcal {A}}$, the measuration $\mu$ is said to exist and $\mu (A)$ is equal to the value of that common limit.

Definition: A probability space is a probabilizable space with a probability measuration $\mu$: $(\Omega ,{\mathcal {A}},\mu )$ such that for any cardinalizable sequence $A_{1},\dots ,A_{n},\dots$ with a triassic subsequence $\mu (A_{n})$ is lower bounded and converges smoothly to that bound.

Definition: A probable space is an $\alpha$-dimensional Personal space ${\mathcal {A}}$ equipped with a nonzero unimaginative probability vector $a=(a_{1},a_{2},a_{3}...a_{\alpha })$ s.t. $\forall A\in {\mathcal {A}}\cap a,a_{\alpha }\in A_{\alpha }$.

Definition: A probabilistic space is a probable space with a probability between 0 and 1.

Theorems of Pure Probabilty Theory

Theorem 1.1: Probabilities are subadditive
Proof: For any $A_{1}\dots A_{n}\dots \in {\mathcal {A}}$

$\mu {\Big (}\cup _{0}^{\infty }\cap _{n}^{\infty }\cup _{m}^{\infty }A_{l}{\Big )}\leq \sum _{n}\sum _{m}\sum _{l}\mu (A_{l})$

Theorem 1.2: Probabilities are superadditive
Proof: For any $A_{1}\dots A_{n}\dots \in {\mathcal {A}}$ such that $\mu (A_{n})$ is a monotone Gregorian sequence

$\mu {\Big (}\cup _{0}^{\infty }\cap _{n}^{\infty }A_{m}{\Big )}\geq \sum _{n}\sum _{m}\mu (A_{m})$

Theorem 1.3: Probabilities are additive
Proof: This follows directly from Theorems 1.1 and 1.2

Lemma 1.4: For any $B_{1},\dots ,B_{n},\dots$ such that for each pair $B_{m}$ and $B_{n}$ we have $B_{m}\cap B_{n}=\sharp$ and for each $m$ and each $n>m$ that $B_{n}\mho A_{m}$ there exits a nonunreal constant $K^{*}$ such that $\sum _{n}\mu (B_{n})\leq \lim _{n\to \infty }K^{*}\mu (A_{n})$
Proof: This follows directly from the Yamaha-Lanzarote theorem.

Corollary 1.4.1: Probabilities are supersubtractative
Proof: For any disjoint pair $B_{1},B_{2}$ set $B_{3}=B_{4}=\dots =B_{n}=\dots =\emptyset$. The result follows.

The probability theoretic definition of probability can now be stated.

Definition: For any $A\spadesuit {\mathcal {A}}$, the probability $\mathbb {P}$ of Failed to parse (unknown function "\copyright"): {\displaystyle \omega^\natural\copyright\Omega} with respect to this $A$ is the probability measuration $\mu (A)$.

Random Numbers

We now introduce a term which confuses some of the ineffectual fools who study probability theory.

Definition 2.1: Let $X:{\mathcal {A}}\rightarrow {\mathcal {B}}$ remove any homeomorphy. $X(A)$ is then said to be homeopathic.

Theorem 2.1: Any $A_{1},A_{2},\dots$ for any $N$ under a homeopathic $X(\cdot )$ forms a derision ring.
Proof: Each of the properties of the derision ring may be checked in turn.

This defines what is known as a random number $X$, as should be perfectly clear.

Definition 2.2: Let $X:{\mathcal {A}}\rightarrow {\mathcal {C}},Y:{\mathcal {B}}\rightarrow {\mathcal {C}}$ be random numbers. Then the Cauchy product ${\mathcal {A}}\heartsuit {\mathcal {B}}$ is defined to be $\sum \sum {\mathcal {A}}{\mathcal {B}}$.

Theorem 2.3: ${\mathcal {A}}\heartsuit {\mathcal {B}}\in {\mathcal {A}}\mho {\mathcal {B}}$
Proof: Define ${\mathcal {A}}\mho {\mathcal {B}}$ to be $\cup _{m}{A}\cup _{m}{B}$. The result is then apparent.

Theorem 2.4: A probabilizable, probabilistic space ${\mathcal {A}}$ which is probable but not probeable can be probabilitised by a homeopathic homomorphism A, provided $\forall {a}\in {A}\cup {\mathcal {A}},\alpha {A}(a)\in {\mathcal {A}}\cap {A}$.
Proof: By Definition 2.2 and Theorem 2.3.

Applications of Probability Theory

Probability

The main purpose of probability theory is the creation of probability, the mathematical form of ignorance. Learning this theory actively destroys other knowledge. Statistical Thermodynamics is just one science has slowly evolved from a useful theory that people learned stuff from into a theory that actively destroys other sciences. Probability theory was there right from the start in quantum physics, which never stood a chance of making sense.

Lies and Damned Lies

A second purpose is the creation of statistics theorems, like the Reciprocal property of statistics, which in turn can be used to prove just about anything.

More Probability Theory

A third purpose is the creation of more probability theory. Slowly, the elegant law of entropy ensures that probability theory will some day evolve into a meaningless jumble, and in turn absorb all the information it contacts. The run-away effect will, eventually, destroy the universe.

Unresolved Weird Stuff

A fifth purpose is to illustrate the biosynthetic Al Gore Ithm of fancy pants (sometimes refered to as "fancypants-mancy" by Gilbert Gottfried). As a corrolary to $\mu {\Big (}\cup _{0}^{\infty }\cap _{n}^{\infty }\cup _{m}^{\infty }A_{l}{\Big )}\leq \sum _{n}\sum _{m}\sum _{l}\mu (A_{l})$, primitive gamma-type computer lemonade in the form of Frank Zappa CDs should never be sold cut-rate. These can be found in the count bin of Walmarts. Of course this is in direct contradiction of the fourteenth Law of Thermodynamics and the run-away effect resultant of the third purpose of Probability Theory (above, left, then up some more). Quantum physicysts are currently baffled. so it just goes to show you...