# Questions tagged [summations]

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### Notation for leads and lags in difference-in-differences

I was hoping someone could help clarify a notational discrepancy. For example, Lord Pischke uses the following sigma notation in two different lecture notes published on the web, yet refers to the ...
29 views

### How to prove absolute summabilities implies the absolute summability of the product series?

In SHUMWAY 2017 Time Series Analysis and Its Applications with R examples 4E, page 486, it states: $\Sigma_{j=-\infty}^{\infty} |a_j| < \infty$ and $\Sigma_{j=-\infty}^{\infty} |b_j| < \infty$ ...
195 views

### Why does absolutely-summable weights ensures a linear series itself summable (convergent)? Some questions on def'n of Linear Series

A "linear series" $y_t$ is the linear combination $$y_t - \mu = \sum_{i=-\infty}^{\infty}\psi_iL^i\nu_t = \sum_{i=-\infty}^{\infty}\psi_i\nu_{t-i}=S(L)\nu_t$$ of weighted (by $\psi_i$ weights) lags ...
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### Summation of two Gaussian distributed data with different coefficient of mean and variance

I need some help on Gaussian distribution. i have two dataset, both are identical and independent distributed, but having mean as 2μ_1 and μ_2, same scenario for the variance. How can I add them? ...
85 views

### Proving an identity involving $E(e_i^2)$ in simple OLS

Once expressed the simple OLS residual $e_i$ as a weighted sum of the noise terms: \begin{equation}e_{i}=\sum_{j}\left(\delta_{i j}-\frac{1}{n}-\left(x_{i}-\overline{x}\right) \frac{x_{j}-\overline{x}...
311 views

### Dot subscript summation notation used in design of experiments

Context I couldn't find a clear explanation of this on this site, and thought that it might be of use. I have provided part of the answer, which anyone is welcome to build from. I'm not sure how to ...
44 views

### Lower incomplete gamma function format in series representation and R [closed]

As known that the lower incomplete gamma function can be written as $\gamma(a,x) = x^{a}e^{-x}\sum_k^\infty{{x^{k}}\over a^{k+1}}.$ What is the format for $\sum_j^\infty{\gamma(v/p-j,rx^{p})}$ in ...
74 views

### How to evaluate a summation equation containing a random variable?

I'm trying to find: $$\Pr(B = 0)$$ Where: $$B = \sum_{i=0}^N b_i$$ And: \begin{align} N &\thicksim \mathrm{Poisson}(\lambda=10) \\ b_i &\thicksim \mathrm{Geometric}(p=0.8) \end{align} ...
35 views

### Summation of an infinite series involving a gamma function, in the context of estimating a Dirichlet prior

I have an unknown multinomial distribution $P^*$ over potentially unbounded set $\Sigma=\{1,2,\ldots,L\}$ from which a training set $\{x^1,\ldots,x^N\}$ has been observed. The observations form the ...
27 views

### Summation of Series involving Exponential terms

I'm currently working on a problem, which involves Poisson-Binomial Distribution. https://en.wikipedia.org/wiki/Poisson_binomial_distribution . The Mean of PBD is given by $M=\sum_{i=1}^{n}p_i$ ....
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### Solving for a difference equation for $s_{t}$
Given $f_{t}=u_{t} - \bar{P}$ and the law of motion for $u_{t} = \rho u_{t-1} + \epsilon_{t}$, where $0<\rho<1$, $\epsilon_{t}$ is mean-zero iid and can be interpreted as a domestic price level ...