Tagged Questions

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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 ...
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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 ...
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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 ...
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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. ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...