Solving problems with graphs

I saw this question on Quora:

A teacher assigns each of her 18 students a different integer from 1 through 18. The teacher forms pairs of study partners by using the rule that the sum of the pair of numbers is a perfect square. Assuming the 9 pairs of students follow this rule, the student assigned which number must be paired with the student assigned the number 1?

A. 16
B. 15
C. 9
D. 8
E. 3

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Markov chain

A Markov chain is a mathematical system that undergoes transitions from one state to another on a state space in a stochastic (random) manner. Examples of Markov chains include the board game snakes and ladders, where each state represents the position of a player on the board and a player moves between states (different positions on the board) by rolling a dice. An important property of Markov chains, called the Markov property, is the memoryless property of the stochastic process. Basically what this means is that the transition between states depends only on the current state and not on the states preceding the current state; in terms of the board game, your next position on the board depends only on where you are currently positioned and not on the sequence of moves that got you there. Another way of thinking about it is that the future is independent of the past, given the present.

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Probability

The fundamental idea of inferential statistics is determining the probability of obtaining the observed data when we assume the null hypothesis is true. For example, if we roll a die 10 times and got 10 sixes, what is the probability of observing this result if we assume the null hypothesis that the die was fair? If the die is fair, the probability of getting 10 sixes in 10 rolls is \frac{1}{6}^{10} = 1.653817e-08, which is a very low probability. Since it's extremely unlikely that we observe 10 sixes on 10 rolls of a fair die by chance, we should reject the null hypothesis. This probability is the p-value.

Let's consider a less extreme case than the previous example. Here I will use the example from the first lecture of the Statistics for Neuroscience (9506) course, whereby a person (the lecturer) claimed that he had to ability to distinguish two different brands of espresso. Our null hypothesis in this case, is that the lecturer doesn't have the ability to distinguish the brands and is simply guessing. We come up with an experiment to test his ability by giving him 8 cups of espresso, where 4 are from brand A and the other 4 are from brand B, and ask him to separate them into two groups. If he managed to correctly group the 8 cups into their respective brands, what is the probability of getting this result if we assume the null hypothesis is true, i.e. what's the probability of getting this result just by chance?

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Combinations and permutations in R

Time to get another concept under my belt, combinations and permutations. While I'm at it, I will examine combinations and permutations in R. As you may recall from school, a combination does not take into account the order, whereas a permutation does. Using the example from my favourite website as of late, mathsisfun.com:

  • A fruit salad is a combination of apples, bananas and grapes, since it's the same fruit salad regardless of the order of fruits
  • To open a safe you need the right order of numbers, thus the code is a permutation

As a matter of fact, a permutation is an ordered combination. There are basically two types of permutations, with repetition (or replacement) and without repetition (without replacement).

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Calculus

I remember studying calculus in school and there were so many concepts that never clicked. I could solve the equations, find derivatives, work out the area under the curve, etc. but I didn't see the use of calculus, i.e. the application of calculus. I'm revisiting calculus now because I've been taking part in a biostatistics course offered freely by Coursera, which requires a working knowledge of calculus. The definition of calculus on Wikipedia is as such:

Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.

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Set notation

I've just started the Mathematical Biostatistics Boot Camp 1 and to help me remember the set notations introduced in the first lecture, I'll include them here:

The sample space, \Omega (upper case omega), is the collection of possible outcomes of an experiment, such as a die roll:

\Omega = \{1, 2, 3, 4, 5, 6\}

An event, say E, is a subset of \Omega, such as the even dice rolls:

E = \{2, 4, 6\}

An elementary or simple event is a particular result of an experiment, such as the roll of 4 (represented as a lowercase omega):

\omega = 4

A null event or the empty set is represented as \emptyset.

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Using LaTeX with WordPress

I've been meaning to learn LaTeX for a while now (among all the other things I want to learn). Here I just illustrate examples of using LaTeX with WordPress, with respect to displaying vectors and matrices.

To display LaTeX, I just need to surround the code with two starting dollar signs (\$\$) and two closing dollar signs (\$\$), without the escapes. Here I just recreate some of the mathematical notation from this tutorial on SVD, as an exercise to learn LaTeX notation and to remember the useful concepts on vectors and matrices explained in the tutorial.

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