Questions tagged [sum]

The sum of two or more random variables.

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17 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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1answer
30 views

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

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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2answers
208 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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25 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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1answer
47 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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1answer
94 views

Expectation of sum of absolute values for correlated normal random variables

Let $x_1, x_2, \dots, x_{N}$ i.i.d. random variables $\sim \mathcal{N}\left(0,\sigma^2_x\right)$. Further, let $z\sim \mathcal{N}\left(0,\sigma^2_z\right)$, $z$ is independent from all $x_i$. We build ...
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18 views

Difference of two independent lognormal random variables by Fourier transform methods?

Is it possible to calculate the difference $X_1-X_2$ of two independent lognormal variables $X_1$ and $X_2$ where $\log(X_1)\sim N(\mu_1,\sigma_1)$ and $\log(X_2)\sim N(\mu_2,\sigma_2)$? Could I ...
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7 views

Is an uneven item number over different facets of a factor a problem?

I have developed a new questionnaire for specifically 1 factor with 4 facets. At first, all 4 facets have the same number of items. I create sum variables and dataframes from them to analyze the ...
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7 views

How to derive this inverse standard deviation error bound for binomial random variable? [duplicate]

Is this error bound suggesting the normal approximation is good for binomial random variables? How was it derived? Why is the reciprocal standard deviation in the error bound?
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1answer
43 views

X~Unif(0, 1) ; X1 + X2 + … X6 = 1 ; Y = sum(X1…X6) ; VAR(Y) =?

Let $X_i$ ~ Unif(0, 1) s.t. $X_1 + X_2 + ... + X_6 = 1$ Let $Y = X_1 + … X_6$ What is $Var(Y)$? (Also the case when it's $X_n$) Purpose for the curious: I'm trying to rank confidence for softmax ...
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52 views

What does “The mean of the sum of N independent variables with the same distribution is N times the mean of a single variable” mean?

I have been reading a book about statistics for physicists and there was this line given: "The mean of the sum of N independent variables with the same distribution is N times the mean of a single ...
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1answer
17 views

Limiting distribution of infinite sparse sum

Let $N$ be a positive integer. I consider $N$ random variables $X_1^{(N)}, X_2^{(N)}, \dots, X_N^{(N)}$, all independent and identically distributed, each taking values $\pm 1$ with probabilities $p/(...
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1answer
33 views

probability distribution of a sum of random variables [closed]

Suppose we have a random variable $X$ $P[X=-1]=1/3$, $P[X=0]=1/3$ and $P[X=1]=1/3$ now let $Y=X^2$ we have $n$ independent realizations of $Y$ $(Y_1, Y_2,......, Y_n)$ what is the probability ...
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1answer
48 views

Distribution sum of correlated normal variables squared

I'm trying to deduce which distribution my data follows and how to estimate the parameters. I have four random variables $X_i \sim N(\mu_i,\sigma_i^2)$ where the means and variances are all different. ...
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110 views

What are continuous distributions that are additive and have finite support

I'm wondering what are continuous distributions that are additive and have finite support. Joint normal distribution is continuous, and is additive in the sense that if $X,Y$ are joint normal, then $X+...
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4answers
524 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
28 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
82 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
42 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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33 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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39 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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55 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
71 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
137 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
61 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
217 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
37 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
106 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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2answers
65 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
157 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
50 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
235 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
157 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
116 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
145 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
31 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
76 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
41 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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1k 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
162 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
199 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
71 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
66 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
169 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
38 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 ...
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1answer
301 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
73 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
90 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 $$ X_n = \sum_{k=1}^{n}Y_k10^{-k} $$ ...