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模倣子 Discrete Normal Distributions

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Introduction

I'm still trying to take it all in, but a buddy of mine clued me in to the idea of Pascal's Triangle producing a distribution, a normal one, and a discrete one, in fact, for an experiment consisting of a series of binomial trials.

His idea is that this discrete distribution may be used to assess the reliability of a poll, for example, taken as a sample from a population about preference for Candidate A versus Candidate B, for instance. I may be misquoting, and I'm still chewing on this idea. He said that this information may be gotten from the "two tails" of the discrete distribution, outside of the probability value of the given sample.

Pascal's Triangle

					0	1	0
				0	1		1	0
			0	1		2		1	0
		0	1		3		3		1	0
	0	1		4		6		4		1	0
0	1		5		10		10		5		1	0

An Example

First off, the fifth row of Pascal's Triangle, as above, is "1, 4, 6, 4, 1" which totals to 16. What does this mean? An experiment of 16 trials, 16 coin flips, will be distributed over no heads and 16 tails to 16 heads and no tails.

It might be simpler to start with the "1, 2, 1" row, the third row. There are four trials, 16 possible permutated outcomes, or 5 combinatorial outcomes.

{ tail, tail, tail, tail}

{ head, tail, tail, tail }

{ head, head, tail, tail }

{ head, head, head, tail }

{ head, head, head, head }

TTTT	TTTH	TTHH	TTHT
THTT	THTH	THHH	THHT
HTTT	HTTH	HTHH	HTHT
HHTT	HHTH	HHHH	HHHT

So the probability of getting one heads flip only is only one in sixteen. The chance of two of each is 50%. 

More Will Be Revealed

I need to get back in touch with my buddy about how to interpret the tails of this distribution, and then how to use it all to assess the likelihood of given results.

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Math Narcolepsy

 During a mathematics course at Columbia University, a student fell asleep in class.

When he woke up, he saw that the professor had written two problems on the whiteboard. He thought they were homework, so he noted them down in his notebook and took them home.

When he tried to solve those problems, he found them incredibly difficult. But he refused to give up. He spent hours in the library, studying through reference books, and finally, he managed to solve one problem, even though it was quite challenging.

In the next class, when the professor did not ask anything about the homework, he was surprised and stood up and asked, "Sir, why did you not ask anything about the assignment given in the last lecture?"

The professor replied, "Assignments? I just wrote them as examples of problems that scientists haven't been able to solve yet."

The student was astonished and said, "But I solved one of them! I even wrote four papers on it." His achievement was later recognized, and all four of his papers are still on display at Columbia University.

The most important thing about this story was that the student didn't hear that "there are no solutions to these problems." He simply recognized that these were difficult questions that needed to be solved, and he tried wholeheartedly to solve them—and succeeded.

This story reminds us – don't listen to those who say you can't do something. Today's generation is often surrounded by pessimism and negativity. Some people deliberately sow the seeds of failure and defeat within others.

But you have the strength to reach your goals, the power to overcome obstacles, and the courage to achieve your dreams. Just believe in yourself – and keep trying.

This student's name was George Dantzig, and this problem was taken from Math Stack Exchange.

"Dantzig proved that in the context of the Student's t-test, the only way we can create a hypothesis test that is independent of the standard deviation is to create a null test, which always rejects or accepts with equal probability, which is not practical."

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