4
votes
2answers
52 views

Approximating the erf function

I was trying to find an approximate solution to the following: $\DeclareMathOperator\erf{erf}$ $$\frac12 \sqrt{\pi} \erf\left (\frac{x-2}{\sqrt{10}}\right) + \frac12 \sqrt{\pi} \erf ...
0
votes
2answers
36 views

What is this pattern?

I have a pattern of numbers from some simulation that I'm trying to piece backwards into some sort of formula. All I can see right now is that they maybe have to do with integrals? The numbers (which ...
-1
votes
0answers
19 views

The summation of product of factorials

So the question is $\sum\limits_{x=0}^n \frac{(\beta+n-x)! (\alpha+x)!}{x!(n-x)!}$. I got the following result from mathematica yet I don't know how to prove it. Can anyone give me some help?
0
votes
0answers
10 views

What is the formula for the maxima in terms of length $n$ of the following modified version of Spearman's footrule distance measure?

Let $\pi$ be a permutation of the natural numbers $1\dots n$ and write $\pi_i$ for the i-th element of that permutation. The following formula computes a modified version of Spearman's footrule ...
0
votes
1answer
109 views

Hypothesis testing for equivalence of two arrangements

I have two arrangements(i.e. permutations) of numbers. First one is the target/real arrangement. Second, is the observed arrangement. e.g. Target := 1,2,3,4,5,6,7 Observed := ...
2
votes
1answer
50 views

Independent sequences

Let $\{x_i\}_{i = 1, ...,n}$ , $\{y_i\}_{i = 1, ...,n}$ be sequences generated by a pseudo-random number generator using different seed keys, for example $ x_0$ and $y_0$. Are $\{x_i\}$ and $\{y_i\}$ ...
1
vote
0answers
108 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
0
votes
1answer
60 views

Expected length of the “greedy” increasing sub-sequence?

Given a sequence of random unique integers of length $n$, if I select every element that is the largest so far how how many elements should I expect to select? This seems superficially similar to ...
0
votes
0answers
55 views

Minimizing association/correlation between two time series

I have two time series, $M_1(t)$ and $M_2(t)$, which can be seen as measurements of two different physical sources, $s_1(t)$ and $s_2(t)$. $M_1$ only depends on $s_1$, whereas $M_2$ depends on both. ...
0
votes
1answer
87 views

infinite sum with each summand converging to zero almost surely

Suppose $X_n$ is a random variable such that $X_n=O(b_n)$ almost surely, with $b_n\to 0$ as $n\to \infty$. Let $C$ be a real constant and $S_{j,k}(X_n)=\sum_{i=j}^kC^iX_n^i$ for ...
0
votes
1answer
46 views

Property of infinite sum

I am looking at the first proof shown here http://mathaa.epfl.ch/cours/PMMI2001/interactive/negbinomexpect_en0.htm and I am a bit lost on how \begin{equation} \sum_{j=0}^{\infty} {j + r \choose r} ...
2
votes
2answers
77 views

Determining Convergence of Power Series

Flip a fair coin until you get the first "head". Let X represent the number of flips before the first head appears. Calculate E[X]. So I solved this problem and you get a power series: $E[X] = ...
2
votes
1answer
44 views

Series involving Marcum Q function

I would like to have a better form of this series: $$\sum_{k=0}^{\infty}\,\frac{1}{k!}\,\left(\frac{ab\sin(c)}{\sqrt{2}}\right)^{2k}\,Q_{k+\frac{3}{2}}\left(ab\cos(c),bx\right)$$ where ...
2
votes
2answers
208 views

Double Summation: Need help to handle $ i \neq j $ : $ \sum_{i=0 \to 7,\ j=1 \to 8,\ i\neq j} (8i + j) $

[Q1]. Can I ? ( write the same summation as ) : $$ \sum_{i=0, i \neq j}^7 \sum_{j=1}^8 (8i + j) \tag{1}$$ I tried to solve the following Summation as follows: Let i = m-1 then, $ \sum_{i=0,\ i ...
1
vote
1answer
47 views

Field of mathematics which deals with similarity of a set of objects each with property variables

(Contextual word of warning: Question written by mathematical novice.) I have a large set of objects. Each object has three variables. Each variable is a number between 0 and 1. For each object in ...
2
votes
1answer
85 views

Behaviour of Two Coupled Sequences Towards a Stable Distribution

The following question arises from research that I am doing in swarm intelligence. The relationships given come from geometric considerations which, I believe, should not be relevant for this problem. ...
3
votes
1answer
65 views

Showing that Poisson sums to 1

The PDF of a Poisson distribution is $P(X=k)=\frac{\lambda^k}{k!}e^\lambda$. As a PDF, it should sum to one; i.e. $\sum_{k=0}^{\infty}\frac{\lambda^k}{k!}e^\lambda=1$. I'm having trouble proving this ...
2
votes
2answers
275 views

Upper bound for the absolute value of an inner product

I am trying to prove the inequality $$ \left|\sum\limits_{i=1}^n a_{i}x_{i} \right| \leq \frac{1}{2}(x_{(n)} - x_{(1)}) \sum\limits_{i=1}^n \left| a_{i} \right| \>,$$ where $x_{(n)} = \max_i x_i$ ...
1
vote
0answers
41 views

Calculating the variance of the a regular expression assuming position dependent

Hello I am having trouble with a slightly biological problem I am given a regular expression $- [RK]-[LV]-[DE]-x(2)-Y$ this expression means that there is a string with the first position being an ...
1
vote
1answer
45 views

How do I know an outlier in a time series isn't the beginning of a trend?

I wish to average detections coming in over time. I use the interquartile range to identify outliers and to discard them. I look at the last 30 values. What do I do if each new value is an outlier, ...
1
vote
0answers
197 views

Calculating step value in range slider using density distribution

I have a javascript range slider with minimum value 0 and maximum value 133K. My initial problem is that this javascript range slider goes up by a step value of 1, meaning that it is relatively ...
1
vote
2answers
437 views

Fast variance calculation

Suppose to have a sequence $X$ of $m$ samples and for each $i^{th}$ sample you want to calculate a local mean $\mu_{X}(i)$ and a local variance $\sigma^2_{X}(i)$ estimation over $n \ll m$ samples of ...
1
vote
0answers
63 views

calculation of the sum using idea of one answer

I am wondering if the sum (the $q$-th moment) in my question Calculation of the moments using Hypergeometric distribution can be calculated using idea in Evaluating 'combinatorial' sum ? ...
1
vote
1answer
111 views

MA process ACF proof - don't understand it

I've got the proof but I don't understand a small detail. As you know for an MA process: $X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$ where $Z_n$ is WGN (pure Gaussian random process). Then the ACF is: ...
2
votes
1answer
309 views

Evaluating 'combinatorial' sum

Help me please to calculate the following sum. I have seen such kind of formulas in the papers related to combinatorics, specifically 'trees'. I am curious how to calculate or approximate this sum: ...
4
votes
1answer
129 views

Calculate $\sum_{i=1}^{[\frac{\sqrt n}{2}]}{n\choose i}$

It is known that $\sum_{i=1}^n {n \choose i}=2^n$. I am wondering what would be the sum if we change the upper limit to $\sqrt n/2$, i. e. How to calculate$$\sum_{i=1}^{[\frac{\sqrt n}{2}]}{n \choose ...
2
votes
1answer
57 views

Closed form for a sequence related to divisibility

If we consider intervals of the form $[1,p_k !! ]$ in which $p_k!! := 2\cdot3\cdots p_k$ we can ask about distribution of primes, near-primes, etc., on such intervals. A naive approach might be to ...
1
vote
2answers
69 views

How can we find a new sum of multiplications based on a previous one?

Suppose wehave two sequences: $$(a_0, a_1, a_2, \dots, a_{2^n-1})$$ $$(b_0, b_1, b_2, \dots, b_{2^n-1})$$ We also have the following sum: $$\sum_{k=0}^{2^n-1}{a_k \cdot b_k}$$ I'd like to know the ...
2
votes
1answer
136 views

Conditions for convergence to symmetric stable distribution?

Under which conditions converges a sum of i.i.d. random variables $$ \frac{1}{a_N} \sum\limits_{n=1}^N X_n $$ to a symmetric stable distribution? Two examples of sufficient conditions are finite ...
1
vote
1answer
410 views

How to calculate/approximate expectation of function of a binomial random variable?

I am stuck at following problem in my research. Suppose that $M=m$ is a random variable with binomial distribution and parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. ...
2
votes
1answer
68 views

Incomplete “round trip” of taking a minimum, then a maximum, from a positively skewed distribution

Let's say you have a distribution that is either symmetric or positively skewed (and defined over 0-1). Call it F. Then, you find the distribution of the minimum of n>1 draws from F. Call it Fmin. ...
1
vote
2answers
215 views

Average, exponential moving average, identities/splitting input parts

I would like to know if certain identities for averages (mean) also hold for the exponential moving average (EMA). I can verify the mean case, but not the exponential case. Can somebody tell me if the ...
1
vote
1answer
54 views

Transform the sample to make it more similar to a given

$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that: Euclidean distance ...
1
vote
0answers
103 views

How to interpret the sum notation on $\sum_{r:i(r)=i} \langle R_{r\alpha} \rangle^{(t)}$

While doing research for my thesis, I ran into a paper called "Statistical Models for Co-occurrence Data". In the early pages, when talking about an iterative numerical method (a custom EM-method, to ...
5
votes
4answers
1k views

Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
0
votes
1answer
318 views

How to calculate correlation between time periods

What would be the best way to calculate some sort of correlation or similarity factor between two lists of time intervals : List1 : 1. 2010-06-06 to 2010-12-12 2. 2010-05-04 to 2010-11-02 3. ...
0
votes
1answer
166 views

Calculating $\sum\left[ x^{2} y \right ]$

Could I calculate the answer to either $$\sum\left[ x^{2} y \right ] - \sum xy$$ or $$\sum \left[ \left ( x - \sum xy \right )^{2} y \right ]$$ If I had any of these variables $ \sum xy $; $ \sum ...
2
votes
3answers
187 views

Determine speed of the object at the current time by the non-uniform time sample

Here is a time sample: $Q = \{(t_i, x_i) | 0 \leq x_i \leq x_{i+1}, 1 \leq i \leq n\}$ and rules: (1) $T_1 \leq t_{i+1} - t_i < T_2$ where $T_1, T_2 > 0$ (2) $x_i$ comes with error: $x_i = ...