Charlie Co

In Part 1 of this video series on calculating the most probable grade histogram, I derived an objective function proportional to the Shannon entropy of the distribution that can be maximized to find the most probable distribution. Now Part 2, goes through the Python coding implementation to solve the constrained maximization problem numerically. Specifically, we solve for the most probable grade histogram for a 50 point test knowing the average grade (40), standard deviation (8.5), and that 20% of the class scored 25 or below.

After tests, students ponder their class grades while teachers seem to design tests that yield bell-shaped grade histograms. In this two-part series, I delve into the math and Python solution for this intriguing problem of finding the most probable distribution. For this video (Part 1), we first derive the number of ways (permutations) N students can be distributed into a grade distribution, noting that the grade distribution with the most number of permutations would be the most probable one. Next, we realize that when the number of students (N) is large, factorials become large quantities that cannot be represented with 64-bit floating point data types. Thus, we turn to deriving Stirling’s approximation (ln N! = N ln N - N), which involves doing the opposite of the traditional Reimann sum. That is, we approximate the sum of rectangular areas by an integral. Following an integration by parts, ignoring small terms, and grouping, we come to a tractable approximation for the natural log of the number of permutations of a grade distribution: -N ∑ p (ln p) where p is the fraction of students with a given score.

Linear regression is typically used to find lines of best fit to data. Here, we transform the equation of a circle to a linear form to fit the center and radius of a circle to data points.

An entirely geometric derivation of the volume formulas for pyramids and cones that offers insight to the origin of \(\frac{1}{3}\) in the volume formulas. This was also my entry to the Khan Academy Junior Breakthrough Challenge.

The rotation matrices are derived here via rotation of the x and y cartesian coordinates and using trigonometric relations to determine the coordinates of a point in the new rotated x' and y' coordinates.

Using the rotation matrices derived earlier, I go through the nitty gritty details of writing Python code, without using image processing packages, to rotate an image of my dad and I pixel by pixel.

Blender models are constructed here to visualize in three dimensions the correspondence in volume of a hemisphere to the volume between a cone and its enclosing cylinder leading to the volume formula for spheres. This landmark discovery was made by Archimedes in the 3rd century BC.

This is the toughest math problem I encountered in high-school. Bringing together the geometric concepts with the multiple absolute value inequalities is quite a challenge.

Archimedes Principle states that the buoyant force is equal to the weight of the fluid displaced. It is used here to derive an expression for the density of a floating cylinder given the density of the fluid, the radius of the cylinder, and the highest point of the cylinder relative to the fluid level.

Most students learn of parabolas as y=ax^2+bx+c. However, parabolas have a more general geometric definition. A parabola is the set of points that are equidistant to a point called the focus and a line called the directrix. Here, the equation of a parabola relative to a point and a line, which is defined by its slope and intercept, is derived. Because the line need not be parallel to the x-axis, the resulting parabola can be oriented at any angle relative to the x and y axes. The resulting equation is implemented and plotted using Python.

This video was made during COVID lockdown when I first started using Python and Google Colaboratory. This was my first video illustrating basic linear regression and plotting with Numpy and Matplotlib.

Animated blender models to illustrate Cavalieri's Principle are constructed step by step in this video using Jacques Lucke incredible Blender add-in Animation Nodes.

A geometric proof of why the centroid of triangles is always located \(\frac{2}{3}\) along any median is explained and illustrated here using a Blender model.

This video goes through the detailed Blender steps in making the figures used in the video Centroid of Triangles.

I used to do YouTube/Twitch live streaming of me playing games. Before I made enough videos to earn a nicer real background, I used to have a green screen, which is replaced with a 3D Blender model image of a wall with a medkit, health potions, and a wall of lighted acoustic tiles. This background is in fact visible behind me in this video, and yes! - that background is 100% fake! This video goes into the step by step details of how the acoustic tile wall section was created using Blender.

Like the earlier video on acoustic tiles, this one goes through the detailed Blender steps of creating the medkit in my background.

This is a fun parody video where I used DaVinci Resolve corner pinning feature to do a screen replacement and create video evidence to convince my friends that I'm playing a healthy dose of video games! DaVinci Resolve is an amazing video editing software provided for free by Blackmagic Design that has been used in editing multiple Hollywood blockbusters.

These simulations are far from accurate as they are not calculated from the so called Navier-Stokes equations. But my gosh, it is so easy and fun to make these fluid simulations in Blender.

The expression for kinetic energy is taught in elementary schools as a simple formula. I dislike memorizing stuff that comes from nowhere, but the traditional derivation of this expression for kinetic energy requires calculus. Here, it is derived using elementary mathematics via conservation of energy.

This video derives an expression for the sum of interior angles of an n-sided polygon. We could not find a suitable background and thought it would be a good idea to record it in a big garbage bag with holes for ventilation and cables for the lamps!

This early video goes through a non-calculus derivation of the formula for circular areas.

I was so happy after making this video that goes through the derivation of sin and cos of 30 and 45 degrees.

This is another early video that talks about the centroid of triangles in a simple way.

This was the first video grandpa commissioned for $30. I miss him. It was the first video we made for what could have been a YouTube channel: Math in a Garbage Bag.