- May 9, 2023
Suppose there are n distinguishable boxes and k indistinguishable balls. Now, we randomly put the balls into the boxes. For each of the boxes, what is the probability that it contains m balls? This is a simple combanitorics problem that can be solved by the stars and bars method. It turns out that in the limit n,k→∞ with k/n fixed, the distribution tends to be a geometric distribution.
- May 1, 2023
For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.
- Mar 30, 2023
For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.
- Feb 6, 2023
Both the Gini coefficient and the variance are measures of statistical dispersion. We are then motivated to find the relationship between them. It turns out that there is a neat mathematical relationship between them.
- Jan 18, 2023
Denote the length distribution of one’s hair to be f(l,t), where l is hair length, and t is time. Considering that each hair may be lost naturally from time to time (there is a probability of λdt for each hair to be lost within time range from t to t+dt) and then restart growing from zero length, how will the length distribution of hair evolve with time? It turns out that we may model it with a first-order PDE.
- Dec 25, 2022
Given your probability of breaking the combo at each note, what is the probability distribution of your max combo in the rhythm game chart? I considered the problem seriously!