Showing posts from September, 2014

Thought Experiment

If you could ask the universe any three yes/no questions, what would they be? I came up with this thought experiment last week while in my introduction to philosophy class. What questions would further our knowledge of science the most? Which questions would benefit society the most? What are the potential consequences of the answers?

In this thought experiment, you can accept the answer to each question as an axiom of the universe. That is, you know the answer is a law of nature. Further, you can assume that a "quantum" answer (i.e., yes and no simultaneously) is not possible.

The first question I would think to ask is Does God Exist?. Humans will never be able to prove or disprove the existence of God, so the answer would give information that we would not be able to find otherwise. The answer to such a question completely change society as we know it (assuming everybody on the planet accepted the answer).

The second question I thought of was Are we alone in the Universe?.…

Simplifying the Expected Value Formula

Probability theory is by far my favorite field in mathematics. I am particularly interested in expected value problems over a discretized domain. I often find myself thinking up and solving expected value problems that may or may not have any actual relevance.

After working on an expected value problem that I had thought up, I derived a very interesting and useful identity for computing the answer to expected value problems more easily in certain situations. My derivation is easiest to understand when the sample space is the set of natural numbers (i.e., $1, 2, 3, \dots$).

In situations when the sample space of a random variable is the set of natural numbers, the expected value is defined by this formula:

$$ \mathbf{E}(X) = \sum_{k=1}^{\infty} k \cdot P(X = k) = P(X = 1) + 2 \cdot P(X = 2) + 3 \cdot P(X = 3) + \dots $$

In some cases, this summation may be difficult to evaluate directly. However, if you represent the sum in a special way, a simple but powerful simplification can be mad…

A Simple Derivation of the Quadratic Formula

In this blog post, I will talk about the famous quadratic formula - a formula for finding the zero(s) of a polynomial equation of degree 2. I first learned this equation in Algebra, but I had always thought of it as a mathematical truth, without actually knowing how it came to be or why it's justified. Now, I have the tools to show that it's correct and to actually derive it using simple algebra.

The quadratic formula is a formula used to solve for the zero(s) an arbitrary polynomial in the form $y = ax^2+bx+c$.  Here it is in it's standard form:
$$ x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a} $$
My proof of this formula relies on an algebraic trick known as completing the square, which is another useful technique to solving equations of this form. While I personally prefer the completing the square method because it's faster, many people prefer using the quadratic formula because it doesn't require much thought or intuition; it's straight plug-and-chug. Here is m…