Differential Equations

A differential equation is an equation that involves derivatives, or differentials. We previously look the differential equation of population growth. \[\frac{dP}{dt} = kP\] or \[P^\prime(t) = k \cdot P(t)\] where \(P\), or \(P(t)\) is the population at time \(t\). This equation is a model of uninhibited population growth. Its solution is the function \[P(t) = P_0e^{kt}\] where the constant \(P_0\) is the size of the population at \(t=0\). In general, differential equations have far reaching applications and have solutions that are functions. …

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Bottleneck in Programming Mathematics

I’ve been trying to retrace my understanding of critical mathematical topics by using programming (Python). Rote calculations and memorization always felt lacking to gain any meaningful insight into mathematics. I expected to be able to quickly code the solutions to any problem so that I could focus on understanding the many mathematical formalities. It turns out that one major bottleneck of trying to learn mathematics by coding are the libraries you choose. …

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Probability: Expected Value & Normal Distribution

Let \(x\) be a continuous random variable over the interval \([a, b]\) with probability density function \(f\). The expected value of \(x\) is defined by \[E(x) = \int_a^b x \cdot f(x)dx\] The Wikipedia’s definition of expected value is useful in this case. The expected value of a discrete random variable is the probability-weighted average of all its possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. …

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How To Convert Investment Returns

Usually, we look at investment returns in terms of annual returns. However, your numbers may be expressed as daily, weekly, or monthly returns. Here’s how to convert them. The idea is that you compound to use the general formula for returns and compound them to an annual period. \[r_a = (1 + r)^n - 1\] Where \(r_a\) is the annual return, \(r\) is the return (in decimal) and \(n\) is the number of periods in a year. …

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Interview with Erik Bernhardsson

I found out about Erik Bernhardsson’s Twitter profile by chance a few years back and was immediately fascinated by his experience and achievements. His website is one of the few that I read on a regular basis. I contacted Erik because I wanted to know more about who he is and his story. He was extremely generous to give me the time of day, given his many responsibilities. I hope you enjoy reading this interview. …

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