Questions tagged [means]
In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. For a data set, refers to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.
1,575 questions
1 vote
2 answers
189 views
Closed form of $P = \prod\limits_{k=1}^{N} k^{k^2}$ [duplicate]
What is the closed form (in terms of $N$) of $\color{blue}{\mathbf{P = \prod\limits_{k=1}^{N} k^{k^2}}}$? When I say closed form, it should not contain another $N$-term or Infinite term product or sum ...
- 2,365
0 votes
1 answer
44 views
Why does the altitude in Courant’s right-triangle figure both equal $\sqrt{xy}$ and lie on the semicircle?
I am studying Introduction to Calculus and Analysis Volume 1. I am reading a section about the Cauchy-Schwarz inequality. I am able to follow the proof of the inequality, thanks to the post(s) How ...
- 267
0 votes
0 answers
23 views
naming family of functions defined by average of exponential functions w.r.t. rates
I have a family of functions defined by "a weighted average of exponential functions across rates from any probability distribution", or mathematically: $$ f(t) = \int e^{-t\lambda} p(\...
- 187
2 votes
3 answers
238 views
Every real polynomial is the mean of two polynomials whose roots are real
PS- in the comments a post has been linked which has a solution but explains no motivation and hence I was not able to understand it and I can't possibly ask the author since they left logn time ago ...
- 3,478
3 votes
1 answer
89 views
If $(a_n), (b_n)$ are positive decreasing sequences, $\sum a_n$ converges and $\sum b_n$ diverges, then the arithmetic mean of $\frac{a_i}{b_i}\to 0$
Inspired by my two previous questions, If $(a_n),\ (b_n)$ are positive decreasing sequences, $(a_n)$ is convex, $\sum a_n$ converges and $\sum b_n$ diverges, then $\frac{a_n}{b_n}\to 0.$ and If $(a_n)...
- 25.2k
0 votes
0 answers
108 views
What's the mean & the variance of a binomial distribution?
The question is as follows: A box contains 2 red and 3 blue balls. Two balls are drawn successively without replacement. If getting ‘a red ball on first draw and a blue ball on second draw’ is ...
- 1
1 vote
1 answer
92 views
Generalizing Means to Points in Euclidean Space
I'm familiar with various classical means for positive real numbers—such as the arithmetic, geometric, harmonic, and, more generally, the power means. For points in the Euclidean plane (or in higher-...
- 13
0 votes
1 answer
150 views
Positivity of a function involving the logarithmic functions
When considering the bounds for the ratio $\frac{B_{2n+1}}{B_{2n}}$ of two Bernoulli numbers $B_{2n}$, I encountered the following problem. For $x>y\ge\exp\frac{1+\sqrt{5}\,}{2}$, \begin{equation*}...
- 2,306
1 vote
1 answer
120 views
If a symmetric matrix $A$ is PSD, then $a_{ij} \leq \sqrt{a_{ii} a_{jj}}$ [duplicate]
In the 3rd lecture of Stephen Boyd's 2023 Stanford EE364A, one student mentions that in positive semidefinite matrices, every entry is less than or equal to the geometric mean of the diagonal entries ...
- 210
1 vote
1 answer
61 views
MSE between sample mean and true mean for vectors
From a population of $N$ vectors I select a sample of $n$ vectors. What will be an error in a sample mean compared with population mean, i.e. how "close" the vectors will be? Does it matter ...
- 2,413
7 votes
2 answers
339 views
Bounds on the probability of $k$-sample mean exceeding the population mean
UPDATE. It turns out that my conjecture is at least as strong as: Manickam-Miklós-Singhi Conjecture. Let $x \in \mathbb{R}^{[n]}$ satisfiy $\sum_{i\in[n]} x_i = 0$. Then, for $k \in (0, \frac{n}{4}] \...
- 186k
1 vote
1 answer
80 views
Can the sample means of a non-integrable random variable be convergent in probability?
I remember reading somewhere that non-integrable real-valued random variables are guaranteed to not fulfil a Strong Law of Large Numbers: the liminf and limsup of the running means are respectively $-\...
- 236
1 vote
0 answers
61 views
Geometric mean for estimating travel times
I want to estimate the expected travel time from some origin O to a destination D. The actual underlying travel time distribution is unknown. However, I have a few historical observations: 20 minutes, ...
- 171
5 votes
0 answers
118 views
4 random points inside the tetrahedron, find the average of the square of the volume at the vertices on these random points
Inside a regular tetrahedron, 4 random points are selected and a new tetrahedron is built on them (let's call it the second tetrahedron). It is necessary to find the average value of the square of ...
- 51
2 votes
1 answer
325 views
Why is circular mean different to arithmetic mean?
The circular mean is calculated as the arithmetic mean of the points on the unit circle corresponding to each angle. As an equation, this is: (from Wikipedia) This works very well for two angles as ...
- 135