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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