Metaphysics on geometric distribution in probability theory

I realized geometric distribution is not exactly about the time needed to get the first success in a given number of trials. This is a very odd feeling. It is probably a feeling applied mathematicians get sometimes, when they feel they are doing the best they can, and yet the theory is not perfect.

This may be a naive post, I warn you, but I was really stunned when I realized this.

Geometric distribution is not about the first success

Let’s jump to the point. We know (or at least, I was taught) that geometric distribution is used to calculate the probability that the first success in k trials (all independent and of probability p) will happen precisely at the k-th trial.

Remember that a geometric distribution is a random variable X such that its distribution is

\Pr(X=k)=(1-p)^{k-1}\,p\,

How can we relate the above distribution with the fact that it matches the first success? Well, we need to have one success, which explains the p at the bottom. Moreover, we want to have just one success, so all other trials must be unsuccessful, which explains the (1-p)^{k-1}.

But hey, where would first ever be written? Continue reading “Metaphysics on geometric distribution in probability theory”

Random variables: what are they and why are they needed?

This article aims at providing some intuition for what random variables are and why random variables are useful and needed in probability theory.

Intuition for random variables

Informally speaking, random variables encode questions about the world in a numerical way.

How many heads can I get if I flip a coin 3 times?

How many people will vote the Democrats at the US presidential elections?

I want to make pizza. What is the possible overall cost of the ingredients, considering all combinations of different brands of them?

These are all examples of random variables. What a random variable does, in plain words, is to take a set of possible world configurations and group them to a number. What I mean when I say world configurations will be clearer soon, when talking about the sample space \Omega (which, appropriately, is also called universe).

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Relationship between reduced rings, radical ideals and nilpotent elements

This post aims at providing some intuition and meaning for the following algebra relationship:

Reduced ring – Radical ideal – Nilpotent

Reduced ring – Radical ideal – Nilpotent

A basic fact of ring theory is that if you take a ring A and quotient it for a (double-sided) radical ideal I you get a reduced ring. Let us suppose A is a commutative ring and understand why this fact is true.

Nilpotent element
Def. a \in A is nilpotent \Leftrightarrow \exists n \in \mathbb{N} : a^n = 0

Informally, a nilpotent element is like a road ending in the middle of nowhere, collapsing in the depth of an abyss. You are driving on it, following the powers of a, and then all of a sudden, with no explanation, your road ends in a big black hole. Indeed, the zero really acts as some kind of black hole, attracting nilpotent-made roads at some point or another: we can think of nilpotent roads as spiraling into the zero.

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Overdetermined and underdetermined systems of equations put simply

This article aims at providing real world examples and links for overdetermined and undetermined systems of equations. Before starting, we will suppose that all over and underdetermined systems are obtained from square systems which admit one and only one solution (i.e. comes from a coefficient matrix with non zero determinant).

Overdetermined systems

When a system of linear equations has more equations than unknowns, we say it is overdetermined. It means what it says: too many rules at once are being imposed, and some of them may be conflicting. Still, it is false to say that an overdetermined system does not have any solutions: it may or it may not.

Intuition

Intuitively, we can think of a system of equations as a set of requests. Imagine you have a group of n people in front of you (the unknowns), and you are supposed to give each person something specific to do. If you give more commands than the number of people, then we have an overdetermined system. It is clear that when this happens, at least one person must have received more than one command. However, there are two possible scenarios.

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Quick method to find line of shortest distance for skew lines

In linear algebra it is sometimes needed to find the equation of the line of shortest distance for two skew lines. What follows is a very quick method of finding that line.

Let’s consider an example. Start with two simple skew lines:

s : \begin{cases} x = 1 + t \\ y = 0 \\ z = -t \end{cases}

r : \begin{cases} x = - k \\ y = k + 2 \\ z = k \end{cases}

(Observation: don’t make the mistake of using the same parameter for both lines. Each lines exist on its own, there’s no link between them, so there’s no reason why they should should be described by the same parameter. If this doesn’t seem convincing, get two lines you know to be intersecting, use the same parameter for both and try to find the intersection point.)

The directional vectors are:

V_{s} = (1, 0, -1), V_{r} = (- 1, 1, 1)

So they clearly aren’t parallel. They aren’t incidental as well, because the only possible intersection point is for y = 0, but when y = 0, r is at (2, 0, -2), which doesn’t belong to s. It does indeed make sense to look for the line of shortest distance between the two, confident that we will find a non-zero result.

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