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 = \...
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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
$k_2 = 1000$ ...
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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 $Y \sim Uniform(c,d)$...
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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:
...
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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 $\...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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?
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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 $x\to +\infty$...
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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 ...
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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 ...
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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 ...
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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 $\sum\limits_{i_1=1}^D\sum\limits_{i_2=1}^{i_1}...\sum\limits_{i_M=1}^{i_{M-1}}w_{i_1i_2...i_M}x_{i_1}x_{i_2}...x_{i_M}$...
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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
...
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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 ...
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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 ...
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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 ...
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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 ...
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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=...
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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+...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 &...
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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 ...
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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.
...
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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 ...
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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 ...
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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&...
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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 ...
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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 ...
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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.
$$\operatorname{Var}\left(\frac{1}{m}\...
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Probability of a sum of random variables falling in a given range
Given a distribution $P$, two values $a$ and $b$, $x=0$, and the following process:
Draw a number $r$ from $P$ and add it to $x$, this is $x = x+r$.
Keep doing this until $x$ is higher than $a$.
...
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Sum of sample given a priori knowledge of its maximum
Given a sample of discrete random variables $X_1, X_2, \ldots, X_n \sim F$, I am looking to calculate the distribution given by the probability mass function:
$$P\left(\sum_{i=1}^n X_i = x~\middle|~\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
user226073
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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 $x_1$...
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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 $\left(f_X(x) = 2e^{-2x}\right)$, ...
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