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Questions tagged [sum]

The sum of two or more random variables.

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Expected value of iid squared conditioned on sum

I would be interested in finding the value of the following expression: $$\mathbb{E}[X_k^2\mid S_N]$$ where $X_k$ are iid random variables with $\mathbb{E}[X_k]=\mu$ and $\operatorname{Var}[X_k]=\...
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Concentration inequality for sums of independent gamma random variables

I am dealing with the following problem: Say $X_1, \ldots, X_n$ are independent Gamma random variables, each one having shape and rate parameters $\alpha_i$ and $\beta_i$, respectively. Let $S_n = \...
HeyCool08's user avatar
4 votes
2 answers
121 views

Confidence interval for the sum of 2 binomially distributed variables

$P_1$ and $P_2$ are uncorrelated, binomially distributed variables with success probabilities $p_1 \neq p_2$. Say I measure: $k_1 = 9$ successes out of $n_1 = 10$ trials for $P_1$ and $k_2 = 1000$ ...
dimitsev's user avatar
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PDF of difference of uniform distributions [duplicate]

Main questions are in bold but feel free to correct me if I'm wrong somewhere else. As far as possible, I need both intuition and formal explanation. Let $X \sim Uniform(a,b)$ and $Y \sim Uniform(c,d)$...
White1Hun's user avatar
1 vote
1 answer
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how to statistically test two sums of 1s [closed]

I have the following vectors: vec_1=c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) vec_2=c(1,1,1,1,1,1,1,1,1) from which I compute the corresponding sums: ...
aaaaa's user avatar
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25 votes
4 answers
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Probability that sum of binary variables is even

Let $S_i \in \{0,1\}$, $i=1,\dots,N$ be $N$ independent random binary variables, each taking the value 1 with probability $0 \le p_i \le 1$ (and the value 0 with probability $1-p_i$). I am interested ...
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An example of a random variable $y\in L^\dagger_2$ having more than one linear combination, $y = \Sigma_{i}\alpha_i x_i = \Sigma_{i}\beta_i x_i$

In the answer for the following exercise: Let $\{x_1,...,x_n\}$ be a finite collection of random variables with $E(x_i^2) \lt \infty$ ($i = 1,..., n$). Show that the set of all linear combinations $\...
Tran Khanh's user avatar
1 vote
1 answer
164 views

Show that for random variable $X$ with $N = \{1, 2, \ldots \}$, $E(X) = \sum_{n = 1}^\infty P(X \geq n)$ [duplicate]

Prove that for random variable with natural numbers from 1 to infinity the expected value $E(X)$ is equal to $\sum_{n = 1}^\infty P(X \geq n)$. Is this the mathematically correct way to prove it? And ...
Ste0l's user avatar
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Contribution of a single value in a Division of Sums

I need to isolate contribution of a single entity in a Division of Sums as shown below. For example, find the contribution of variable a in the following: (a 1 + b ...
Maddy's user avatar
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The third central moment of a sum of two independent random variables

Is it true that in probability theory the third central moment of a sum of two independent random variables is equal to the sum of the third central moments of the two separate variables?
AdVen's user avatar
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1 vote
1 answer
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How to deal with a summation term in a regression model?

In the following fixed-effects model, $EI$ is a dummy variable indicating an economic integration agreement in place between $i$ and $j$. $A$ is used to index the specific agreement an $i, j$ pair ...
ametricsb's user avatar
7 votes
3 answers
936 views

Questions about Wilcoxon signed rank test

I wanted to conduct a Wilxocon signed rank test but stumbeld upon two questions that I am unable to solve on my own. I tested 2 types of interfaces for a software with the same ten people. I want to ...
Lukas Pezzei's user avatar
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1 answer
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Why is the distribution of the sum of the values on two dice bell-shaped and symmetric if two uniform dist is triangular distribution?

Why is the distribution of the sum of the values on two dice bell-shaped and symmetric if two uniform dist. sum is triangular distribution via Irwin-hall distribution?
jkj's user avatar
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5 votes
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If $Z=X+Y$, and I know the probability distribution of $Z$ and $Y$, and $X\perp Y$ how to recover the probability distribution of X?

Suppose I know the distribution of $Z$ and $Y$: $Z\sim F_Z$ with density $f_Z$, $Y\sim F_Y$ with density $f_Y$. Suppose I also know that $Z=X+Y$, where $X$ and $Y$ are independent and the ...
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Non-negative fat-tailed "almost stable" family of distribution with finite mean?

I am looking for a finite-dimensional family of distributions $F_X(x)$ with all the following properties: Supported on $[0, +\infty)$, Fat tailed, i.e. $(1-F_X(x)) \sim x^{-\alpha}$ for $x\to +\infty$...
AndreA's user avatar
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Estimating the distribution of a sum of two random variables if the family of one of the variables is known

Assume I have a random variable $Y=X_1+X_2$. I want to estimate the distribution $f$ of $Y$ given a sample $y_1,\ldots,y_N$. If this was all that is known about $Y$ the best way would probably be to ...
LiKao's user avatar
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2 votes
1 answer
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Estimating the probability of a sum of events

I have n machines that use the same utility. Each machine randomly demands a unique f_n flow rate of the utility once every h_n hours on average. Each machine's demand event lasts for about m_n ...
Nathan Bevan's user avatar
12 votes
7 answers
2k views

Noise cancels but variance sums - contradiction?

I have been told both things with regard to e.g. summing noisy time series, to justify opposing expectations. On the one hand, I have been told to expect that summing multiple noisy inputs should lead ...
benxyzzy's user avatar
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1 answer
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How to rewrite a multivariate polynomial term without redundancies?

This is from an exercise found here on page 60. My question is: what is meant by $\sum\limits_{i_1=1}^D\sum\limits_{i_2=1}^{i_1}...\sum\limits_{i_M=1}^{i_{M-1}}w_{i_1i_2...i_M}x_{i_1}x_{i_2}...x_{i_M}$...
user1169080's user avatar
1 vote
0 answers
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Summing multiple standard deviations (repeated measures) [duplicate]

Say a set of 5 participants completed two subtests (A and B). The scores on these subtests can be summed to get a total test score. Here is the dummy data: Participant Subtest A Subtest B Total test ...
Alice's user avatar
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3 votes
2 answers
457 views

Model Sum of Squares from ANOVA table

Background: I am studying a course on statistical experiments using the textbook by Douglas Montgomery on the analysis of experiments. This is an introductory course and so I am relatively new to the ...
FD_bfa's user avatar
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Given a set of random variables, how can I find a linear combination of these variables satisfying a constraint on the sum of their permuations?

Say I have n random variables, {X0...Xn}, n>9. I also have another set of random variables constructed from the first set, where each of these are the sum of 9 ...
Eli Katz's user avatar
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92 views

How to add noise into a standard distribution without increasing its variance?

Suppose I have a standard distribution dataset X with a mean 0 and std 1. Now I want to create slight variations of this data by injecting some noise. I could make ...
Anonymous's user avatar
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What is difference between the joint probability distribution and the sum/convolution, of 2 dists? [duplicate]

Google is coming up a bit short when I searched for "joint vs sum random variables". Perhaps someone can provide an authoritative answer to compare and contrast the sum/convolution of 2 ...
Alexander Mills's user avatar
2 votes
0 answers
90 views

Decomposing the prediction of a sum of Gaussian Processes into predictions from each Gaussian Process

Suppose the functions $f_1\sim\mathcal{GP}(m_1,K_1)$ and $f_2\sim\mathcal{GP}(m_2,K_2)$ are drawn from independent Gaussian Processes, and let $$f=f_1+f_2.$$ Then $$f\sim\mathcal{GP}(m,K)$$ where $m=...
FizzleDizzle's user avatar
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Average of random variables from two Poisson distributions?

I'm lost with a very simple question of finding the average of random variables from two Poisson distributions. I know that if $X\backsim Pois(L1)$ and $Y\backsim Pois(L2)$, then $X+Y\backsim Pois(L1+...
jvkloc's user avatar
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1 answer
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Data wrangling in R with dplyr: How do I consolidate rows? United States Census data [closed]

I am analysing population projection data from the United States census and I need to present population estimates by race/ethnicity for each year from 2020 to 2029. The US census separates Asian-...
Jack Murphy's user avatar
4 votes
1 answer
63 views

Convergence of an infinite sum of weighted independent and identically distributed random variable

Let $z_i$ be $i.i.d$ random variable with $E(z_i)=0$ and $E(Z_i^2)=1$ with a symmetric distribution. Further, $|\beta|<1$. Now consider $\sum\limits_{i=1}^{\infty} \beta^i (z_i+|z_i|)$. I want to ...
mathstat's user avatar
1 vote
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142 views

How can I estimate the sum of coefficients

I am trying to estimate the cumulative effect. When I have an ols regression with many dummies as explanatory variables, can I sum the coefficients to find the cumulative effect? If yes, how do I find ...
eva's user avatar
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251 views

Sum of dependent multivariate gaussians

Note: I have already seen this Wikipedia article, and similar questions on this website: 1. Given two dependent multivariate Gaussian random variables, is the sum also a multivariate Gaussian? $X \sim ...
muser's user avatar
  • 145
3 votes
1 answer
203 views

Conditional probabilities of the parameters

I have the following function $$ x(k) = \sum_{m}^{M} e^{i(U_m k + \beta_m)} $$ Where $$ i = \sqrt{-1} $$ The $U_m$ values come from a normal distribution and the $\beta_m$ values come from a uniform ...
CfourPiO's user avatar
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Time series benchmarking/reconciliation and revisions - are there methods that minimise revisions?

I am using the tempdisagg R package for benchmarking quarterly time series to annual time series from different (more trusted) sources (by temporally disaggragating the annual data using the quarterly ...
SiKiHe's user avatar
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0 answers
24 views

Does $p=0 \implies \sum_{i=1}^{p} \phi_i L^i = 0$?

Let us take this $\operatorname{AR}(p)$ equation $$\left(1 - \sum_{i=1}^{p} \phi_i L^i \right)X_t = \mu + \epsilon_t$$ as an example. When $p=0$ I read this to mean \begin{align*} \mu + \epsilon_t &...
Galen's user avatar
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1 vote
1 answer
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Is there a clear interpretation of Corr(X, X+Y) in research?

Consider a case of $Corr(X,Z)$, often found to be high; where later, it was found that it holds exactly $Z = X + Y$. In effect, the previously found correlations were equal to $Corr(X, X+Y)$. How can ...
amc____'s user avatar
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7 votes
3 answers
2k views

If X=Y+Z, Is it ever useful to regress X on Y?

If we have X and Y that are mathematically dependent: X = Y + Z, is it 'forbidden' to use Y as a predictor to X in linear regression? I'm trying to find a concise explanation for why it is, or isn't. ...
amc____'s user avatar
  • 85
9 votes
3 answers
1k views

Is the sum of two singular covariance matrices also singular?

I have two sample covariance matrices, computed from $n$ samples, less than $p$ variables: they are singular then. I know that the sum of two covariance matrices is also a covariance matrix. My ...
Larel5000's user avatar
2 votes
1 answer
58 views

Prove $P(X_1+X_2> 2C) \leq P(X_1>C)$ if $X_1,X_2$ are identical, but dependent?

If $X_1,X_2$ are dependent but identically distributed, it seems obvious that $P(X_1+X_2\geq2C) \leq P(X_1\geq C)=P(X_2\geq C)$. At least if we additionally assume that the joint distribution is ...
Jome's user avatar
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0 votes
3 answers
597 views

Statistical Data Analysis using "Sum" Function

Most commonly when I hear descriptive data analysis using statistics these following functions are often inclded: Mean Standard Deviation Variance Range Mode Median etc. Is the function "Sum&...
user963241's user avatar
1 vote
1 answer
126 views

Calculating the probability of the total duration of N sequential events with different cdfs describing their duration

Be patient, I am not very skilled with cdf. I seem to have a seemingly simple problem for which I either can't seem to find material about or simply lack the vocabulary for. Given are N sequential ...
Pepijn Ekelmans's user avatar
2 votes
1 answer
426 views

Distribution of sum of $n$ random variables with mixture of two exponential distributions

Suppose that the random variable $Y$ follows a mixture of two exponential distributions, that is \begin{equation} f_Y(y) = \sum_{i=1}^{2}\pi_i f(y| \lambda_i) \end{equation} where $\pi$ stands for ...
Statistics 's user avatar
2 votes
1 answer
323 views

R function to compute variance of average of correlated random variables

I want to calculate the variance of the average of n correlated variables. I found a formula for that in Borenstein et al. (2009) Introduction to Meta-Analysis. $$\operatorname{Var}\left(\frac{1}{m}\...
Dr Ljotsson's user avatar
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0 answers
142 views

Probability of a sum of random variables falling in a given range

Given a distribution $P$, two values $a$ and $b$, $x=0$, and the following process: Draw a number $r$ from $P$ and add it to $x$, this is $x = x+r$. Keep doing this until $x$ is higher than $a$. ...
alexmolas's user avatar
  • 288
8 votes
2 answers
412 views

Sum of sample given a priori knowledge of its maximum

Given a sample of discrete random variables $X_1, X_2, \ldots, X_n \sim F$, I am looking to calculate the distribution given by the probability mass function: $$P\left(\sum_{i=1}^n X_i = x~\middle|~\...
StephenSwat's user avatar
2 votes
1 answer
186 views

sumscores instead of factorscores or SEM

Suppose I would like to use sumscores after running a confirmatory factor analysis (CFA) with two latent factors. The items for each factor are then summed and in subsequent analyses these sums are ...
BenP's user avatar
  • 1,838
4 votes
3 answers
139 views

Sum of Discrete Uniforms, but each value can be picked no more than N times?

Suppose there are i.i.d. variables $X_{1,..n}$ with discrete uniform distribution with the support $[1, n]$. What would be the distribution of such a sum if we introduce the condition that each value ...
Igor Yegin's user avatar
2 votes
1 answer
91 views

For k independent variables, if each one is independent of $Y_1$,...,$Y_p$, how to formally prove their sum is also independent of each $Y_p$?

SUppose I have $X_1,...,X_k$ independent of each other. I also have $Y_1,...,Y_p$ is independent of each other. If each one in $X_1$,...,$X_k$ is independent of each one in $Y_1$,...,$Y_p$, how to ...
ExcitedSnail's user avatar
  • 2,966
2 votes
0 answers
81 views

When is $\sum Z_i \sim \sqrt{n} Z_i$?

If $X_i$ are independently and identically distributed $N(0,\sigma^2)$ then $Y=\sum X_i \sim N(0,n\sigma^2)$, i.e. $\sum X_i \sim \sqrt{n}X_i$. That raises two questions: Is a zero-mean normal ...
Henry's user avatar
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-1 votes
1 answer
200 views

Consistency when we want to find the distribution of sum of random variables following each one a distribution

I want to clarify a point that disturbs me among different cases. I am interested in formulate correctly in a general case when we know the distribution of different random variables and we want to ...
user avatar
0 votes
0 answers
26 views

How do I sample Simultaneous Sums of Gamma-Distributed Variables?

Suppose I have 7 variables $y_i$ sampled from $Gamm(a,1)$, with $a>0$. Now, I define $$x_1 = y_1+y_2+y_3+y_4,$$ $$x_2 = y_1+y_2+y_5+y_6,$$ $$x_3 = y_1+y_3+y_5+y_7$$ What is the distribution of $x_1$...
Arthur Campello's user avatar
1 vote
1 answer
77 views

Probability of joint dependent events

I'm having trouble finding a way to do this calculation and checking if I'm correct: Let $X_1 \sim Exp(2)$ and $X_2 \sim Exp(2)$ be independent random variables $\left(f_X(x) = 2e^{-2x}\right)$, ...
Ander's user avatar
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