Questions tagged [sum]

The sum of two or more random variables.

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9 views

How to get the solution of a combination of a summatory function with multiplication of vectors of numbers (general problem)? [closed]

I am trying to solve the following mathematical equation: 4[8∑q1(1 - q1)*q2(1-q2)] q1 = is a vector of length 10000, with values between 0-1 q2 = is a vector of length 10000, with values between 0-...
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4answers
417 views

How to interpret sum of two random variables that cross domains?

suppose we have two discrete random variables: $X: \{$6 sided dice rolls$\}$ $\rightarrow \{1..6\}$ (following uniform distribution) $Y: \{$coin flips$\}$ $\rightarrow \{0,1\}$ (following uniform ...
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1answer
15 views

Perform signed rank test where there are duplicates in data?

How do you perform a rank test when there are duplicates in the data? that is, we have a dataset with numbers $1,1,1,1,3,3,3,3,4,4,4,4.$ and another dataset also with duplicates.... Is the sum of ...
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1answer
59 views

How do I find the conditional distribution of a normal r. v. z, given that I know the sum of z and another normal r. v. x is greater than some value?

Suppose I have two independent normal random variables, $X$ and $Z$ with $\mu_x$, $\sigma^2_x$ and $\mu_z$, $\sigma^2_z$. Suppose I also know that $x+z\geq y$. How do I find the conditional ...
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1answer
37 views

Why is $\sum{(x_i-\overline{x})^2}$ = $\sum{(x_i-\overline{x})x_i}$ true? [duplicate]

I have seen this equality many times in books but I never found an explanatory derivation.
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5 views

Infering noise conributions on the sum of normal RV's

Suppose multiple factors affect the noise in a measurement, e.g. a manufacturer may have some variance between production runs ($\sigma_1^2$), and some variance between products within the same ...
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0answers
31 views

When (if ever) is the sum of two dependent geometric RVs negative binominal?

Imagine you have two random variables $X $ and $Y$, you know $$ X \sim \text{Geometric}(p) \\ X + Y \sim \text{Negative Binomial}(2, p) $$ I am interested in what if anything can be said about the ...
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34 views

Independence of random variables and sums of random variables

I am seeking to find the joint distribution of X and Y. I have the marginal distributions of X and X+Y and they are independent. We have that $f(X=x,Y=y)=f(X=x,X+Y=x+y)$ which is equal to $f(X=x)f(X+...
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0answers
25 views

Convolutions of joint random variables

I have two discrete dependent random variables $X,Y$, where both $X$ and $Y$ can take values either $0$ or $1$. Furthermore, I know their joint distribution $f_{X,Y}(X,Y)$. Now let's say I have an ...
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2answers
57 views

Why is the sum of all the elements in a Gaussian-distributed list with zero mean not zero?

If I generate a list of elements which has a Gaussian distribution with zero mean: List = np.random.normal(0, 1, 500) my intuition (why is obviously wrong) tells ...
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1answer
47 views

Aggregation with an overlap: Dirichlet distribution

Suppose that we have $$(p_1,p_2,p_3,p_4)\sim Dirichlet(a_1,a_2,a_3,a_4),$$ where $p_4=1-p_1-p_2-p_3.$ When we add random variables for example, $p_1+p_2$ and $p_3+p_4$, the resulting distributions ...
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1answer
59 views

residuals in the simple regression model

The residuals in the simple regression model have to sum up to 0?
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1answer
88 views

Linear regression $y_i=\beta_0 + \beta_1x_i + \epsilon_i$ covariance between $\bar{y}$ and $\hat{\beta}_1$

I am currently reading through slides from Georgia Tech on linear regression and came across a section that has confused me. It states for $$ y_i=\beta_0+\beta_1x_i+\epsilon_i $$ where $\epsilon_i \...
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2answers
36 views

How can I prove these propositions of infinite sum of random variables?

$x_1, x_2, x_3, ..., x_i, ...$ ~ $uniform(0, 1)$ The actual random variable is the following. $P_i = (1-x_1)(1-x_2)...(1-x_{i-1})x_i$ And the goal is proving these... $\sum_{i=1}^{n}P_i \leq 1$ If ...
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1answer
73 views

Does the sum of discrete uniforms coverge to a discrete Gaussian?

Is there some analogous of the Central limit theorem for discrete uniforms and discrete normal distributions? To be more specific, let's say we have identical and independent random random variables $...
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24 views

Dealing with multiple independent variables with the sum of Linear Regressions

Suppose I want to predict a quantity (in week $t$): $V_t := \sum\limits_{i=1}^{10} V_{i,t}$. We do this by (simple) linear regression on each of the individual quantities that make up the sum, with ...
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1answer
51 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
102 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
48 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
158 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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2answers
97 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 ...
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1answer
101 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
104 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
21 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
32 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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2answers
39 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
807 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
152 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
193 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
53 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
51 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
103 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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1answer
37 views

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 ...
7
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1answer
254 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
67 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
35 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
82 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
7 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
112 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
55 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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19 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
62 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
20 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
514 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
286 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
269 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
132 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 ...
7
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1answer
1k 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
52 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 ...