Paradigm Shift: All numbers are random variables.
Aug 6, 2025
I've recently had a paradigm shift, that has bridged 'probability' (which I usually think of as a foreign sort of math) and 'regular math' (alebgra, calculus, etc). All the numbers we deal with in 'regular math' (be it constants or coefficients like 2 and 3.79), can all be thought of as RANDOM variables (just like how we consider x,y, and z to be variables). It's just that these real numbers/constants are typically "deterministic" variables, i.e. we know their values with complete certainty. But say you had an equation y= 2x+3 (you're trying to model some linear relationship) and you got those coefficient values (2 and 3) from your friend. Say this friend is not the most trustworthy guy. Then, your coefficients are actually "probabalistic" variables, and not "deterministic" variables. Although the distribution from which these coefficients were drawn from may appear to be truly random (because your friend's mind is infinitely random), there are many numbers in nature that are actually drawn from bona fide probability distributions (normal, poisson, beta)... That's where the fun comes in. Because what happens when you do calculus with
What about x and y and z, you ask? Well they can be drawn from distributions too. It's just that their values are literally not determined yet. But if you know which distribution they come from (say x is from a normal distribution), then you can have a better idea of how the equation behaves - because although the graph/function is valid over the whole domain of real numbers, the x value is from a distribution that is centered around a mean, and so you know in what regions of the graph y is more likely to be found. (Still thinking about y=2x+3). So then you can ask, what determines the valueu of the mean of x's distribution? I think that comes from historical data/domain knowledge. You can also ask, for what functions will the output variable still be of the same distribution as the input variable. Like for \( y =\frac{x^2}{sin(x)} \), will y be from a normal distribution?
AHA Moments from Linear Algebra (3rd time around)
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coming soon...
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coming soon...
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Life at Dartmouth: A Sophomore's Perspective
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Reflecting on my sophomore year at Dartmouth, I'm amazed by how much I've grown both academically and personally. The combination of rigorous coursework in mathematics and computer science, along with the vibrant campus community, has shaped my perspective in ways I never expected...