Questions tagged [summations]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
1answer
18 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 $\Sigma_{j=-\infty}^{\infty} |b_j| < \infty$ ...
0
votes
1answer
46 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 ...
0
votes
1answer
43 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? ...
0
votes
1answer
77 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}...
0
votes
0answers
44 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 ...
2
votes
1answer
36 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 ...
1
vote
1answer
71 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} ...
1
vote
0answers
33 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 ...
2
votes
0answers
24 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 $M=\sum_{i=1}^{n}p_i$ ....
0
votes
1answer
21 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^...
3
votes
0answers
39 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{...
1
vote
0answers
84 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 ...
0
votes
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 ...
2
votes
2answers
80 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,...
1
vote
0answers
69 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 $$\sum e_i\hat{Y}_i = e^T \hat{Y} = ((I-H)Y)^T HY$$ ...
2
votes
1answer
87 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 ...
0
votes
1answer
248 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)...
5
votes
1answer
81 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 ...
0
votes
1answer
2k 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.
2
votes
1answer
80 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{...
0
votes
1answer
94 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 ...
3
votes
1answer
54 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 ...
0
votes
2answers
1k 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 ...
1
vote
1answer
343 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 ...
1
vote
1answer
40 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,...
0
votes
1answer
23 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 ...
3
votes
1answer
567 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}^...
0
votes
1answer
118 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 ...
0
votes
0answers
28 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 $$\mathbb{P}([X=x \cap Y= z-x] \bigcup ... \bigcup [X=x+z \cap Y= z]) = \sum_{x=0}^z \mathbb{P}(X=x \cap Y=z-x)$$ ?
5
votes
2answers
255 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 ...
0
votes
1answer
199 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 ...
4
votes
2answers
168 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 ...
4
votes
1answer
125 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 ...
0
votes
1answer
202 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 ...
2
votes
1answer
775 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 $\...
-1
votes
1answer
14k 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-...
13
votes
1answer
3k 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 ...
2
votes
1answer
1k 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.
5
votes
1answer
78 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 ...
2
votes
0answers
33 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 ...
0
votes
0answers
411 views

Calculate variance of subpopulation given variance of overall population and its complement

I'd like to be able to calculate the sample variance for a subpopulation [B] given the sample mean and variance for its complement [A] and the overall population [A+B]. I have the sample mean and ...
1
vote
2answers
154 views

Sum of combination

My problem is: Evaluate: $$\sum_{i=0}^n i{n \choose i}$$ I only know that $$\sum_{i=0}^n{n \choose i} = 2^n$$ not so sure when an "i" is added. What is the step of this evaluation?
4
votes
1answer
126 views

Sum of products in an expected value

A box contains $n$ balls numbered from 1 to $n$. Suppose you take a ball at a time, putting it back on the box, until you pick a ball twice. How many balls are you expected to take from the box? Let $...
2
votes
0answers
598 views

Summation of a product

I need to calculate the following expression: $$\sum_{k=1}^N a_k b_k$$ ${a_k}$ and $b_k$ are real positive numbers. N and k are integers. I know the average values of $a_k$ , defined as $\overline {...
1
vote
0answers
168 views

variance of summation/compound variable?

here is my situation. I am weighting a packet of material that has 10 individual units in it. In the end of the day I would like to know the average weight and variance of the individual units but the ...
32
votes
4answers
16k views

Generic sum of Gamma random variables

I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ...
1
vote
0answers
73 views

Random variables for which the distribution of the sum of the RV with a Gaussian RV is known

For a counter example, I am searching for random variables $Y$ such that for a independent normal random variable $X$ the distribution of $Z=Y+X$ is known parametrically. Ideally, the Shannon entropy ...
1
vote
1answer
70 views

Why is this MGF identity true?

If $X_i \overset{i.i.d.}\sim N(\mu, \sigma^2) $, we know that: $\bar{X} \sim N(\mu, \sigma^2 /n)$. But why does: $$\exp\left({\sigma^{2}\over 2}\sum_{i=1}^{n}(t_{i}-\bar{t})^{2}\right)= M_{X_{1}-\bar{...
4
votes
1answer
175 views

What can we conclude about the distribution of the sum of two random variables?

If we know, for independent random variables $X$ and $Y$, $P(X>x)\leq0.05$, and $P(Y>y)\leq0.05$, can we say anything about $P(X+Y>x+y)$? Can we be certain that it is less than $0.05$? Under ...
0
votes
1answer
416 views

How to rewrite a sum of probabilities formula as multiplications?

I have an equation like that: $p(r|s)= \frac{p(s,r)}{p(s)}=\frac{ \sum_{w,c} p(c,s,r,w)}{\sum_{w,c,r} p(c,s,r,w)} $ I am new to probability and I want to learn that how can I write sum formula as ...