Articles of average

Continuously sampled event: Estimating the value of a future data point, based on past measurements and their tendency

Problem I’d appreciate some ideas on how to define a formula to estimate the value of a future data point for a continuously sampled event, based on past measurements and their tendency. At any given time, I have exactly 15 past measurements of the event. Let’s assume that what I’m trying to predict is the […]

Very fascinating probability game about maximising greed?

Two people play a mathematical game. Each person chooses a number between 1 and 100 inclusive, with both numbers revealed at the same time. The person who has a smaller number will keep their number value while the person who has a larger number will halve their number value. Disregard any draws. For example, if […]

Prove inequality of generalized means

Consider the generalized (power) mean of positive numbers $a_1, \dotsc, a_n$ $$M_p(a_1, \dotsc, a_n)=\left(\frac{a_1^p + \dotsb + a_n^p}{n}\right)^{1/p}\qquad p\in \mathbb{R}$$ where for $p=0$ we use the geometric mean. The generalized mean inequality says that $$ p < q \implies M_p(a_1, \dotsc, a_n) \leq M_q(a_1, \dotsc, a_n),$$ with equality holding iff $a_1 = \dotsb = a_n$. […]

Sum of averages vs average of sums

I have essentially a table of numbers — a time series of measurements. Each row in the table has 5 values for the 5 different categories, and a sum row for the total of all categories. If I take the average of each column and sum the averages together, should it equal the average of […]

Mean value of the rotation angle is 126.5°

In the paper “Applications of Quaternions to Computation with Rotations” by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by integrating quaternions over the 3-sphere). How can I make sense of this result? If rotation angle around a given axis can be […]

Average bus waiting time

My friends and I were “thinking” yesterday in the pub about the following: if a person is standing on a bus stop that is served by a single bus which comes every p minutes, we would expect the average waiting time to be p/2 (which may or may not be correct). But we had no […]

Asymptotic difference between a function and its binomial average

The origin of this question is the identity $$\sum_{k=0}^n \binom{n}{k} H_k = 2^n \left(H_n – \sum_{k=1}^n \frac{1}{k 2^k}\right),$$ where $H_n$ is the $n$th harmonic number. Dividing by $2^n$, we have $$2^{-n} \sum_{k=0}^n \binom{n}{k} H_k = H_n – \sum_{k=1}^n \frac{1}{k 2^k}.$$ The sum on the left can now be interpreted as a weighted average of the […]

3 trams are coming every 10, 15 and 15 minutes. On average, how long do I have to wait for any tram to come?

3 trams are coming to the stop every 10, 15 and 15 minutes. On average, how long do I have to wait for any tram to come? It’s a practical problem, not some kind of a riddle for which I have a surprising magic trick or an answer. I really don’t know. I was waiting […]

Average $lcm(a,b)$, $ 1\le a \le b \le n$, and asymptotic behavior

What is the average value for $\mathrm{lcm}(a,b)$, with $ 1\le a \le b \le n$, for a given $n$, and what is the asymptotic behavior? The $\mathrm{lcm}$ is the least common multiple. I have calculated, as $(n, avg)$: $$(10, 19.836)$$ $$(100, 1826.859)$$ $$(1000, 182828.976)$$ The values appear to converge quadratically to some constant. This generalizes […]

Geometric mean of prime gaps?

The arithmetic mean of prime gaps around $x$ is $\ln(x)$. What is the geometric mean of prime gaps around $x$ ? Does that strongly depend on the conjectures about the smallest and largest gap such as Cramer’s conjecture or the twin prime conjecture ?