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Questions tagged [sum]

The sum of two or more random variables.

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1answer
35 views

Inferring random variables from their sum

Suppose I have a large set of receipts that list the items I bought, but only list the total cost. One day I might have bought Milk, Butter, and Eggs. A different day I might have bought Bread, Milk,...
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2answers
51 views

Deconvolution of the sum of three gaussian distributions

Consider the sum of three normal random variables: $ R_{i,j}=A_{i}+B_{j}+C_{i,j}\, $ where $ A_{i}∼N(μ_{A},σ_{A}) $ , $ B_{j}∼N(μ_{B},σ_{B}) $ and $ C_{i,j}∼N(μ_{C},σ_{C}) $ . Assuming $A$, $B$ ...
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1answer
40 views

Force sum of random varables to equal to 1 [duplicate]

Suppose I have 3 random variables, $X1, X2,X3$. Define $Z$ as: $Z=X1+X2+X3$ I want to force $Z$ to equal 1 for every "realization" of $X1,X2,X3$ ($X_i \sim Beta(a_i,b_i))$. As an example, let $X_i$ ...
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1answer
37 views

Understanding the infinite sum of random variables

I am doing a course on time series analysis, and am struggling with this definition: We call a weakly stationary process $\{X_t\}$ invertible with respect to a white noise $\{\epsilon_t\}$ if ...
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1answer
51 views

How to check these sequences generated by i.i.d random variables are martingales?

Let $\{Y_n\}_{n\geq 1}$ be a sequence of independent, identically distributed random variables. $P(Y_i=1)=P(Y_i=-1)=\frac12$ Set $S_0=0$ and $S_n=Y_1+...+Y_n$ if $n\geq 1$ I want to check if the ...
5
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1answer
67 views

Is the sum of trends of two time series the trend of the sum of the time series?

Let's say I have two time-series, A and B. I build time-series C as C=A+B. I estimate the trend of A, let's say I get +0.5 (Theil-Sen). I estimate the trend of B, let's say I get -0.4 (Theil-Sen). ...
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1answer
58 views

Mean of root-sum-square of

Suppose that I have several normally-distributed random variables xi, each with its own different variance. All x's are zero-mean and independent. If y is the root-sum-square of the xi's, how do I ...
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1answer
13 views

Confusion about the derivation of the TD-Learning update rule

I am currently trying to understand the paper "Learning to Predict by the Methods of Temporal Differences" by Sutton. I am stuck with the following step: (From "Learning to Predict by the Methods of ...
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1answer
17 views

Covariance of sums of pairs of correlated variables

Take two vectors of normally-distributed random variables $\mathbf{x} = (x_1, x_2, \ldots x_n)$ $\mathbf{y} = (y_1, y_2, \ldots y_n)$ where the covariance of each pair $(x_i, y_i)$ is known, $\...
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0answers
12 views

Sum of multivariate lognormals

Is it possible to approximate the sum of multivariate lognormals using Wilkinson approximation? Any reference?
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2answers
37 views

Is the error term a sum of r.v.?

`If in a econometric model I have: $y = \beta x + u$ where u is the error term, we have: $u = y - \beta x$ Supposing that $\beta=1$, $y\sim N(0,1)$, $x \sim N(0,1)$ and $x$, $y$ are independent. ...
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0answers
26 views

Normal random variables arithmetics?

Given two normal random variables : N1(mean1,std1,count1) and N2(mean2,std2,count2). Where 'count' is the number of values used to build N1 and N2. BTW I don't have access to the values used to create ...
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0answers
251 views

Comparing sum of values of two groups

I have a question about how to make a comparison between the aggregate values of two experimental groups. Say that in an experiment with a control group and a treatment I collect the number of times ...
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0answers
143 views

Probability of sum of sequences of integers

Let K be a positive integer.Suppose that the integers 1,2,3,...,3k+1are written down in random order.What is the probability that at no time during this process, the sum of the integers that have been ...
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1answer
182 views

How could “sum of exponential distribution is 1” be proven?

$$f(x; \lambda) = \begin{cases} \lambda e^{-\lambda x} \quad \text { for } x \geq 0 \\ 0 \quad \quad \quad \text { for }x < 0\end{cases} $$ How can I prove that the sum of probabilities under ...
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2answers
43 views

Summation of squared x_i if summation of x_i is 1

How to prove "If $\sum_{i=1}^n x_i=1$, then $\sum_{i=1}^n x_i^2>1/n$"? I'm thinking about $Var(x_i)=E(x_i^2)-[E(x_i)]^2=\frac{1}{n}\sum_{i=1}^n x_i^2-1/n^2\ge0$. Is that correct?
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2answers
39 views

Covariance of random variables whose sum is less than a constant

Suppose that we have integer random variables $X>0$ and $Y>0$ and constant number $a$. We have: $X+Y < a$. Can we say that the covariance of these random variables is less than or equal to ...
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1answer
68 views

Sum of 2 Normally Distributed Random Variables With a Correlation

I've been given a problem where I have $$ X \sim \mathcal{N}(\mu = 2, \sigma^2 = 9) $$ $$ Y \sim \mathcal{N}(\mu = 3, \sigma^2 = 4) $$ Their correlation is $ \rho_{XY} = 0.6 $. First I am asked for ...
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Summing up double series under constraints on the indexes

I have the following double sum: $$ \sum_{t=0}^\infty \sum_{\ell=0}^r \psi(t,\ell,r), $$ only for even values of $t+\ell$ or $t+\ell=0$. First, I thought, since $\ell$ depends on $r$, and $r$ can ...
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1answer
227 views

Generate identically distributed dependent normal random numbers with prespecified sum

How do I generate $n$ identically distributed but not independent normal random numbers such that their sum falls within a prespecified interval $[a,b]$ with probability $p$? (This question is ...
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1answer
61 views

Question regarding the distribution of sum of random variables

Let $X_1, ... X_n$ be i.i.d random variables that have an exponential distribution with parameter $\theta$. So we know that $\sum X_n \sim \Gamma(n, \theta)$. This makes sense by working backward. ...
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0answers
33 views

property about the standard deviation of 2 r.v.

If $X,Y$ are $\geq 0$ random variables, how to demonstrate that: $$2*Stdev(X) \leq Stdev(X+Y)+ Stdev(X-Y) $$ $Stdev$ represents the usual standard deviation.
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1answer
60 views

Convergence of sum of exponentially weighted random variables

I don't know if the title is accurate, but I have this problem: I have iid RVs $Y_k$ that has a value from {0,1,...,9} with equal probability. I need to show that $$ X_n = \sum_{k=1}^{n}Y_k10^{-k} $$ ...
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0answers
6 views

Sum of values by different groups (2 joined DBs)

I’ m facing an issue getting the results for sum of values by specific group of data from 2 combined databases. Both DBs contain same attribute called ProductID. First DB in includes column category ...
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0answers
71 views

Finite sum of beta prime iid random variables

The beta prime distribution is infinitely divisible, as proved in Steutel and van Harn, 2003 (Appendix B). Sadly, in this book, there is no espression of the parameters of the distribution of n ...
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1answer
50 views

Decomposition of the probability of the sum

I cannot understand how is gotten the following decomposition. Supposing that $X_1,...,X_n$ random variables i.i.d with heavy tailed distribution $S_n=\sum_{i=1}^nX_i$ In the article that I m ...
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0answers
16 views

Calculating minimum sum of ordered statistics

Can someone help me with this problem? Thanks and apologies in advance! The following sample of observations of X is given: 1,2,2,3,4,4,5. Calculate
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1answer
52 views

ANOVA & Sums of Squares in Excel and R

In this simple dataset, I have bird populations from three watersheds. I am completing an ANOVA by hand and in R and comparing the outputs. ...
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0answers
211 views

Approximation of an infinite sum in R [closed]

I'm calculating the probability mass function for a count variable and the normalization term is an infinite sum of the form $\sum_{n = 0}^{\infty} f(n)$. I'm looking for a function in R that ...
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19 views

simplification of below sum

Is it possible to rearrange/simplify the sum to obtain an expression for k in terms of a: $\sum_{x=k}^{\infty}{x+n-1\choose x}\theta^n(1-\theta)^x = a $
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1answer
29 views

Is it possible to simplify the below summation?

I was doing some calculations, but cannot proceed further without some simplification to the below summation, is it possible to simplify it, so that it doesn't involve a summation? $\sum_{x=0}^{s}\...
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2answers
468 views

Sum of predicted values to the power of 10 [closed]

When I take predicted values from a linear model to the power of 10, their sum is always a lot bigger than the original. Is it even allowed to sum, and does anybody have a reference for how this ...
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0answers
207 views

Linear combination of non central chi-squared random variables

I want to analyze the distribution of $$X = \sum_i X_i^2$$ where independent $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$. If $\mu_i=0$, I can derive the distribution by passing Gamma distribution like ...
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1answer
183 views

Correlation between Weighted Sum of Random Variables and Individual Random Variables

Given the following set of random variables and constants, $\newcommand{\inreala}[2]{\in \mathbb{R}^{#1 \times #2}} \newcommand{\var}{\mathrm{Var}} \newcommand{\cov}{\mathrm{Cov}} \newcommand{\corr}{\...
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0answers
91 views

Propagation possion errors on scaled count bins

I have a count-channel histogram, where the counts have a standard Poisson uncertainty - if bin $i$ has $C_i$ counts then the uncertainty is $\sqrt{C_i}$. Now if I were to sum all the bins my job ...
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1answer
807 views

Sum of forecasts

I have a question regarding forecast. I'm building an inventory model around warehouses, where all warehouses have multiple customers/countries assigned. I have data on sales for all countries ...
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0answers
45 views

Distribution of future value of equal cash-flows with log-normal returns

If I'm making equal payments $C$ each period into an investment with log-normal one period returns such that $r_i \sim N(\mu, \sigma)$, is there a simple way to derive the distribution of future ...
7
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2answers
178 views

Probability of k zeros give the sum of n Poisson random variables is t?

Suppose that I have $X_1,X_2,X_3,...X_n$ iid random variables from a Poisson distribution of parameter $\lambda$. Given that $X_1 +X_2+X_3 +...+X_n = t$, what is the probability that exactly $k$ of $...
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0answers
615 views

Distribution of the sample mean of Poisson random variables

Suppose that you have data x which is modeled as a realization of a Poisson random variable X with expected value $\lambda$>0. I know that the sum of Poisson random variables is also Poisson ...
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0answers
133 views

a single estimate and the population mean

There is a physical parameter $Z$ defined as $Z=\frac{X}{X+Y}$, where $X=\sum_{i=1}^{N} x_i$ and $Y=\sum_{i=1}^{N} y_i$. I want to find an interval that brackets the population mean $\mu_Z$ for a ...
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0answers
83 views

Standard error for sum of random variables

I am conducting an analysis of pharmacokinetics data in which I have plasma concentration of drug for different dosage groups. In this way, the area under the concentration-dose curve is computed as $...
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1answer
107 views

Sum of k most extreme values

I have $n$ balls, which I put independently and at random into $\ell$ bins or urns. I then look at the $k$ bins with the most balls inside and count the total number $S$ of balls in these bins. What ...
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1answer
101 views

How do I test for trend in sum of random variable?

I have a dataset with cost measurements for hospital admissions of ~25,000 patients over multiple years. Cost seems to be log-normal distributed and taking the logarithm results in an approximately ...
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0answers
395 views

Square roots of sums absolute values of i.i.d. random variables with zero mean

In an earlier question, I asked about the limiting distribution of the square root of the absolute value of the sum of $n$ i.i.d. random variables each with finite non-zero mean $\mu$ and variance $\...
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0answers
50 views

to calculate the distribution of m*(A+B)^2+n*abs(A-B)

Assume that $A$ and $B$ are two independent random variables both following a normal distribution with a zero mean and a variance $\sigma^2$. How to calculate the distribution of $m \cdot (A+B)^2+n \...
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1answer
80 views

Can we show this sum of Gamma CDF converges, and if so can we derive its limit?

This is a bit of a strange question, but suppose I have some random variables. $$Y_i \sim Gamma(i,\lambda)$$ Where this comes from the fact that each $Y_i$ is defined as the sum of $i$ independent ...
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0answers
21 views

Efficient computation of the value-at-risk of a credit portfolio via summation of independent discrete random variables

With the application of computing a value-at-risk for a credit portfolio (tail percentile of a distribution) in mind, I am currently contemplating the possibility of estimating the distribution tail ...
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1answer
121 views

Particular sum of i.i.d. random variable

I am computing the following probability: $\Pr \left[ \sum_{j} (P G_j R_j) \le T\right]$, where $P$ is a fixed number, $G_j$ is a random variable i.i.d. $\forall j$ and $R_j$ is a term that depends ...
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1answer
441 views

Sum of random variables without normalization approaches gaussian

The central limit theorem states that the limiting distribution of a centered and normalized sum of independent random variables with mean $\mu$ and finite variance $\sigma^2$ is Gaussian. $$ \frac{\...
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1answer
1k views

Finding the distribution of sum of Lognormal Random Variables

I am trying to find the distribution of sum of 2 lognormal random variables. I referred the literature available on Cross validated, Stack overflow and few papers before posting this. I used ...