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Questions tagged [summations]

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10 views

Operator ranking of sum and plus

In the book I am reading on page 308 I find the following formula: My question is which operator has the higher rank - the sum or the plus? In other words: How would I correctly set the brackets in ...
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1answer
43 views

Summation notation

I am reading a statistics book which says: " If $ X \sim N ( \mu, \sigma^2)$, it is verified that: $ \sum_{i=1}^{n}X \sim N ( n\mu, n\sigma^2) $ My doubt is if it should have been written as $ ...
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51 views

Is there a way to prove $\mathbf{\hat{Y}}^T\mathbf{e}=\mathbf{0}$ without resorting to summations?

I would like to show that $\mathbf{\hat{Y}}^T\mathbf{e}=\mathbf{0}$. I can solve this by saying that it is equivalent to showing $\sum e_i\hat{y}_i=0$. However, I'm wondering if there is a way to ...
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1answer
173 views

Probability of compound Poisson process

Let $X$ be a compound Poisson process with rate $\lambda$ and increments $Y_i = \pm 1$ with probability $\frac{1}{2}$. Find $P(X(t) = 0)$. I tried conditioning on $N(t)$: $$ P(X(t) = 0) = P(\sum\...
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1answer
95 views

How to evaluate Probability of Y?

Hi all, It's my first undergraduate statistics module as a business major and I've encountered some difficulties in computing the response to the question. I have several queries below: Would Y have ...
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1answer
140 views

Summation in a Network using identity activation function

I am currently experimenting with a network with one input-layer, one hidden layer, and one output-layer. I am using the identity-function as the activation-function. During the forward-pass, i began ...
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19 views

Sum of pure errors

We know that in a simple linear regression model that the sum of all the residuals is 0 but why is it that the sum of all the pure errors is also 0? Is there a relation between them?
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1answer
29 views

Making a discrete probability question continuous

I'm trying to figure out how many coin flips you'd need to have a greater than 50% chance of having seen a heads, given a biased coin with heads probability $p$. From this question we can see that ...
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1answer
48 views

Value of $\sum_{j=1} (y_{j} - \bar{y})$ and proving properties of hat value

The i-th fitted value $\hat{Z}_i$ is written as a linear amalgam of response values $\hat{Z}_i=\sum_{j=1}h_{ij}Z_j$ where $h_{ij}=\frac{1}{n}+\frac{(y_i-\bar{y})(y_j-\bar{y})}{S_{yy}}$ and $S_{yy}=\...
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1answer
1k views

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 ...
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1answer
34 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 $\Sigma_{j=-\infty}^{\infty} |b_j| < \infty$ ...
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1answer
260 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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1answer
79 views

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? ...
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1answer
93 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}...
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639 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 ...
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1answer
52 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 ...
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1answer
78 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} ...
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37 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 ...
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31 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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1answer
22 views

How to explain the following discrepency when changing parameter for exponential?

Suppose $y$~ $\exp(\frac{1}{2\theta})$ then $\frac1\theta y$~$\exp(\frac12)=\frac14\chi^2_2=\frac14\gamma(1,2)$ then $\sum \frac1\theta y=\frac14\chi^2_{2n}$ then $\sum y$~$ \frac{\theta}{4} \chi^...
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55 views

Summation of combinations up to $r-1$ terms

I am trying to come up with a simplified expression for $$\sum_{k=r}^{n}\binom{n}{k}$$ Choosing $x=y=1$ in Binomial theorem, I have $$2^n = \sum_{k=0}^{n}\binom{n}{k}$$ $$2^n = \sum_{k=0}^{r-1}\binom{...
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108 views

How does this manipulation of summation of exponents work?

I found this paper by Fischer and Igel (2012) (1). I am not able to get past this last step in their derivations (eqn 22, p24) where $h_i$ are values of $h$. Can anyone tell me how they reduced the ...
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1answer
2k views

how to Differentiate the kNN formula?

I am going through a book on statistical learning and ran into a problem concerning k nearest neighbor methods. The book says that using least squares to determine optimal k will lead to $k=1$. I ...
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2answers
97 views

Summation of the Matrix of a Squared Sum

I find that: $$\Big(\sum_{i = 1}^n y_i\Big)^2 = \sum_{i=1}^m y_i^2 + \sum^m_{i\neq j}y_iy_j$$ where $m=(n^2-n)=n(n-1)$, the first part of the right side is the sum of the main diagonal of the matrix,...
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86 views

How to show that the summation of the product of residuals and fitted values equal 0?

I would like to show that $\sum e_i\hat{Y}_i$ = 0 with $\hat{Y} = HY$, $H$ - hat matrix and $e$ - the residuals $(I-H)Y$. So far, I've gotten $$\sum e_i\hat{Y}_i = e^T \hat{Y} = ((I-H)Y)^T HY$$ ...
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1answer
168 views

Aggregation of distributions: Independent vs. perfectly rank correlated

Suppose $X_1, X_2,\ldots,X_n$ are random variables distributed according to p.d.f's $p_{X_1},p_{X_2},\ldots,p_{X_n}$ (most of which are extremely right-skewed). We randomly sample $S$ values from each ...
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1answer
514 views

What is the probability density function (pdf) of the dot product of M complex normal random variables?

What would be the probability density function (pdf) of the complex random variable given below? $$Z = \sum_{i=1}^{M}{x_{i}^{*}y_{i}}$$ where $x_i, y_i$ are independent r.v.'s with $\mathcal{CN}(0,c)...
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1answer
105 views

Summation of Two random variable

Suppose $X$ is random variable with PDF $f(X)=2(x-1)$, $1 \le x \le 2$; $Y$ is a random variable with a triangle pdf with minimum at $2$, mode at $2.5$, and maximum at $3$. Is it possible to define ...
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1answer
6k views

Regression proof for decomposition of sums of squares [duplicate]

I got as far as distributing the summation across the Left Side so that I have: $$ \sum_i y_i^2 - \sum_i 2 y_i \bar{y} + \sum_i \bar{y}^2 $$ Not sure where to go from there.
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90 views

Sum of function of normally distributed values

I have a given number, $v$, given by $$v = \sum_{i=1}^N\left\{v_i \cdot \left[m \cdot \left(1 - b_i \cdot \mathbb{1}_{\geq w}(v_i)\right) - b_i \cdot \mathbb{1}_{< w}(v_i) \right] - a \cdot \mathbb{...
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1answer
134 views

Is there such thing called sum of absolute sum?

Sum of absolute difference (SAD) is common in image processing as a measure of similarity. Is there such thing called sum of absolute sum in image processing or other field? Does it have any ...
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1answer
61 views

Proving a property of $(n-1)s^2$

I would appreciate your help as I climb the stats learning curve! I want to prove the following: "Let $x_1, x_2, ... , x_n$ be any numbers and let $\overline x = (x_1 + x_2 + ... + x_n)/n$ Then ...
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2answers
4k views

What is the variance of the sum of Yi's

Seems a simple enough question, and I presume that, if Yi are normally distributed, Var(Sum(Yi)) = Sum(Var(Yi)) This feels like I'm jumping to the wrong conclusion though. Any help would be ...
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1answer
484 views

Formative/composite variable as weighted additive index

I'm very new to SEM (and relatively new to stats generally). I would like to ask whether my plans generally sound reasonable before I get lost for days in the literature. The model that I try to work ...
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1answer
54 views

Characteristic function issue

As mentioned in a previous post, I've been trying to work through ALL of the problems in Jacod and Protter's Probability Essentials. The following problem has been giving me issues: Let $Z \sim N(0,...
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1answer
26 views

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 ...
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1answer
999 views

Prove that sum of uniform distribution (-1,1) is also uniform (-n,n)? [duplicate]

If $d_i \in U(-1,1)$ (uniform distribution between -1 and 1 - not sure what the canonical notation is for this), then it seems intuitive that $\sum_{i=1}^n d_i \in U(-n,n)$ and thus $$P\big(\sum_{i=1}^...
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2answers
258 views

Which is the right way to apply PCA on different sized matrices

I am working on human age classification where I have four descriptors, namely GEI, FED, UC ...
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33 views

Probability of union of disjoint events $[X=x \cap Y= z-x]$ for $x=0,1,...,z$ equals $\sum_{x=0}^z \mathbb{P}(X=x \cap Y=z-x)$

Why can one write $$\mathbb{P}([X=x \cap Y= z-x] \bigcup ... \bigcup [X=x+z \cap Y= z]) = \sum_{x=0}^z \mathbb{P}(X=x \cap Y=z-x)$$ ?
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358 views

Sum of random variables without central limit theorem

I know that using central limit theorem we approximate sum of random variables into Gaussian distribution. Is the any other approximation method available for finding the probability distribution ...
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1answer
250 views

Help with proof of factorization criterion

In a proof of the factorization criterion regarding sufficient statistics I came across the following derivation: Consider the set $ A_s=[(y_1,..,y_n:s(y_1,..,y_n)=s] $ Now somewhere along the only if ...
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179 views

Distribution of sum of function of two random variables

Let $\{x_1, \ldots, x_n\}$ be a set of $n$ i.i.d. samples from a distribution $p(x)$. I would like to evaluate the distribution of the sum $$ S = \sum_{1\leq i<j\leq n} f(x_i, x_j), $$ where $f$ is ...
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1answer
129 views

Using Poisson distribution to evaluate summations

I'm interested in how to use a Poisson distribution to evaluate $\sum\limits_{x=0}^\infty \frac{(x^2-x+1)(2^x)}{x!}$ I see that this is similar to the general pmf form of $\frac{2^{x}}{x!}$. My ...
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1answer
359 views

Variance of a Sum

I've got the following random variable for which I must find the expected value and variance: $X_h =\sum_{i=1}^{15} X_i$ Where $X_i$ is a random variable of the set $s = \{0, 1, 3\}$, corresponding to ...
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1answer
1k views

Measuring share contribution of each var/cov term to the standard deviation of a sum of variables

Say, for a simple example, I have a random variable $X = \alpha_1 X_1 + \alpha_2 X_2$, where $X_i$ are random variables and $\alpha_i$ are weights. I then calculate the standard deviation of $X$ as $\...
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18k views

In SPSS, should I calculate the mean score or the total sum score?

Are there any differences between mean scores and total sum scores? I know how to calculate both, I just don't know if there is one of them that is preferable? Also, the scores will be used for t-...
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1answer
4k views

What does it mean if the median or average of sums is greater than sum of those of addends?

I'm analyzing the distribution of network latency. The median upload time (U) is 0.5s. The median download (D) time is 2s. However, the median total time (for each data point, T = U + D) is 4s. What ...
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1answer
2k views

Expectation of cube of summation of independent random variables

Where would I begin on this problem? I know I begin with pulling $c^3$. Where would I go from there? And I know that $\mathbb{E}[X] = x_1p_1 + .... x_n p_n$ I'm stuck on the rest, however.
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1answer
93 views

multi stage binomial "process"

I wish to model the retransmission time of a file that divided into K blocks. I know the successful blocks of first transmission obey the binomial distribution $$ X_1 \sim \text B(K,p) $$ , p is the ...
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38 views

Standard deviation of number of terms in a sum

If some random variables are drawn from a normal distribution N(m, s) with m > 0 until the sum of the draws exceeds some ...