Questions tagged [sum]

The sum of two or more random variables.

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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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2 votes
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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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1 vote
1 answer
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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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2 votes
1 answer
157 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 ...
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2 votes
1 answer
81 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}\...
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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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8 votes
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368 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|~\...
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Meaning of set notation applied in time series analysis proof

Let $\{ Y_{t}\}$ be a stationary process with autocovariance function $\gamma_{k}$. Let $\bar{Y} = \frac{1}{n}\sum_{t = 1}^{n}Y_{t}$. Show that: $Var(\bar{Y}) = \frac{\gamma_{0}}{n} + \frac{2}{n} \...
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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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2 votes
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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 ...
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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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1 vote
1 answer
32 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 $\left(f_X(x) = 2e^{-2x}\right)$, ...
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2 votes
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Numerical evaluation of infinite sums

I am working with Skellam random variables and I would like to evaluate the CDF of the absolute value of a Skellam random variable in which both Poisson random variables have the same rate, $\lambda_1 ...
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How is a convex combination of Dirichlet-distributed variables distributed?

Let $X = (X_1, \dots, X_K) \sim \operatorname{Dir}(\alpha_1, \dots, \alpha_K)$ and define the convex combination $Y = \sum_{i=1}^{K} c_i X_i$. In the case of $K=2$, the constraint $\sum_{i=1}^{K} X_i =...
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How to apply Lyapunov CLT to data

I have a situation where I have around 30 classes of variables with different means and variances (though the means aren't too far from eachother; think 4-7) and that the distributions are right ...
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1 vote
1 answer
208 views

2 approaches for Monte-Carlo : weighted sum of $\chi^2$ distribution and Moschopoulos distribution with Gamma distribution

If I take as definition of $a_{lm}$ following a normal distribution with mean equal to zero and $C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})$, and if I have a sum of $\chi^2$, can I write the 2 ...
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1 vote
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55 views

Probability that any element of a random unit-length vector is large [closed]

Given a vector $X \in R^n = \{x_1, x_2, ..., x_n\}$ drawn uniformly such that: $x_i \in [0, 1]$ for all $i$; and $\sum x_i = 1$, how would you find the probability that any of the $x_i > y$, for ...
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If $20 $ random numbers are selected independently from the interval $(0,1) $ probability that the sum of these numbers is at least $8$? [closed]

If $20 $ random numbers are selected independently from the interval $(0,1) $ what is the probability that the sum of these numbers is at least $8$? I tried to take this question https://math....
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Is the sum of 3 bits a linearly separable task?

In other words can a linear classifier learn to correctly assign a class (label 0 to 3) for an input of 3 bits? Intuitively this cannot work, since the half-adder circuit contains an XOR block, which ...
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Sum of a number of shifted exponentially distributed random variables

I know that the sum of $k$ independent exponentially distributed random variables each with density function: $$\displaystyle \lambda\,{{\rm e}^{-\lambda\,x}}$$ has an Erlang distribution: $$\...
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1 vote
1 answer
151 views

Expectation of Maximum and Minimum of Partial Sums of Normal Random Variables

Peggy Strait, 1974, Pacific Journal of Mathematics ON THE MAXIMUM AND MINIMUM OF PARTIAL SUMS OF RANDOM VARIABLES Gives a nice result (4.3) and (4.4) in terms of "standard normal random variables&...
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6 votes
4 answers
260 views

What is an intuitive explanation for Q90 (X+Y) > Q90(X) + Q90(Y) in fat-tailed variables. Non Subadditivity

In a business situation, management keeps a reserve of money for a 'rainy day' just in case costs are more than expected. The 90th percentile ($Q_{90}$ in the following) might be an indicator of how ...
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72 views

Concentration of sum of geometric random variables taken to a power

I am interested in techniques for showing the concentration of sum of $n$ iid geometric random variables $X_1, X_2, \cdots, X_n$ (number of trials until success), say with success probability $p = 1/2$...
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Operator ranking of sum and plus

In the book I am reading on page 308 I find the following formula: My question is which operator has the higher rank - the sum or the plus? In other words: How would I correctly set the brackets in ...
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105 views

Conditional expectation, conditional on sum of weighted average of two iid RVs

I have an arbitrary distribution $F$, and two variables $z, x \sim F$. I only observe the weighted average $y = \alpha z + (1 - \alpha) x$. Conditional on $y$, what is the expected value of $z$? I ...
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3 votes
0 answers
44 views

Mean number of throws to exceed a threshold [duplicate]

Say that you have a die with n faces, and you need to throw the die until the sum of your results exceeds a given threshold. What is the average number of throws needed? I think that to compute that ...
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1 vote
1 answer
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Sum of estimated costs for uncertain events

I have a number of possible events $e$ with a probability $p_e$ of the event occuring and a cost estimate should the event occur (if it doesn't occur the cost is 0). The probability for each event is ...
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1 vote
1 answer
48 views

Summation notation

I am reading a statistics book which says: " If $ X \sim N ( \mu, \sigma^2)$, it is verified that: $ \sum_{i=1}^{n}X \sim N ( n\mu, n\sigma^2) $ My doubt is if it should have been written as $ ...
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3 votes
1 answer
109 views

What is the expectation of $\left\langle (n \bar{y})^4 \right\rangle$, if $y_i \sim \mathcal{N}(\mu,\sigma^2)$?

Let $y_i \sim \mathcal{N}(\mu,\sigma^2), \; i = 1,\ldots,n$ and $\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i$, such that $n \bar{y} = y_1 + \ldots + y_n$. Then, we want to know what the expectation of $(n \...
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Fast Evaluation of a Double Sum

Let $q$ be a probability distribution on $\mathcal{X}$, $w$ be a nonnegative function from $\mathcal{X}$ to $\mathbf{R}$ which is bounded away from $0$ and $\infty$, and $s$ be a bounded function ...
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2 votes
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How can we decompose $\text{Var}[\sum_{i=1}^n\sum_{j=1}^m f(A_i,B_j)]$?

Formulas for decomposing the variance of a summation of random variables can be found on Wikipedia but what is the variance of a double summation of a function of random variables? That is, are there ...
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Is there a way to prove $\mathbf{\hat{Y}}^T\mathbf{e}=\mathbf{0}$ without resorting to summations?

I would like to show that $\mathbf{\hat{Y}}^T\mathbf{e}=\mathbf{0}$. I can solve this by saying that it is equivalent to showing $\sum e_i\hat{y}_i=0$. However, I'm wondering if there is a way to ...
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2 votes
1 answer
289 views

Probability of compound Poisson process

Let $X$ be a compound Poisson process with rate $\lambda$ and increments $Y_i = \pm 1$ with probability $\frac{1}{2}$. Find $P(X(t) = 0)$. I tried conditioning on $N(t)$: $$ P(X(t) = 0) = P(\sum\...
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1 vote
0 answers
41 views

Sum of Bimodal Distributions? [closed]

If I'm trying to estimate a the sum of a bunch of random variables, where each random variable is a bimodal distribution, how would i go about thinking or modeling what that looks like?
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1 vote
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Z-Score for Sum of Proportions?

First, please bear with me. I am not very savvy with statistics, so I may be mixing up terms or using things improperly. Second, I am dealing with statistics related to baseball, so let me explain ...
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170 views

Sum of IID normal variables with index following Poisson distribution

$X_1, X_2,\ldots$ are a sequence of independent normal random variables with mean 1 and variance 1. Calculate the variance of $X_1+X_2+X_3+\ldots+X_{N+1}$ where $N$ follows Poisson distribution with ...
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539 views

What statistical test can compare the sum instead of the mean?

I'm confused about if t-test can only test means (sum / sample size), or if it can test sums as well (not normalizing for sample size). Below is a passage from a trusted book. Note that even though ...
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2 votes
1 answer
123 views

Proportion of the sums or average proportion of the parts?

Here’s a curiosity that come up during a work discussion. While this example uses financial data, it has a statistical question at its heart. Consider the following table with a budget and actual ...
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-1 votes
1 answer
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I need to find the Z-score of a population to use as a cutoff point in order to reduce the value sum of numbers to a new sum [closed]

This may be tough to describe, but I'll give it a shot. I am setting up an analysis that produces a large set of numbers. Let's call this Analysis 1 (A1). If I run A1 and return 60 numbers with a ...
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5 votes
2 answers
567 views

Is there a statistical distribution whose values are bounded $[-1,1]$ and sum to 1?

The Dirichlet distribution contains values that are bounded $[0,1]\in \mathbb{R}$ and sum to $1$. Is there a parametric distribution or similar method whose values do the same but reach as low as $-1$?...
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  • 3,089
4 votes
0 answers
128 views

Any known approximations of summing quantiles from joint (bernoulli / lognormal) distributions

This is my first post to this site! For an insurance-like scenario, I have several independent risks which I want to sum together and find a 95% percentile. Currently I do this by Monte Carlo but I ...
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  • 201
3 votes
1 answer
148 views

How to calculate this dependent probability (marbles without replacement)?

I present the question in two steps: First: Let there be 100 bags. A person puts 5 marbles into 5 separate, randomly selected, bags. You are now to collect the contents of the bags, one by one. If you ...
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