Questions tagged [sum]

The sum of two or more random variables.

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7 views

How to sum values in a row (and not in a column) in Excel? [closed]

I would like to calculate the statistical error of my times T average. I would like to use the following formula where $T_i$ represents the separate values for the measured times. I know that I can ...
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23 views

Variance of ratio of sums

How do I compute the variance of a ratio of sums ? $$ Var\big(\frac{\sum_i X_i}{\sum_j Y_j}\big) $$ I have 2 datasets $X=(X_1,...,X_n)$ with $Y=(Y_1,...,Y_n)$ that I need to compare, and estimate the ...
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1answer
35 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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23 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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1answer
13 views

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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25 views

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

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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1answer
44 views

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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1answer
29 views

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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20 views

Sum of Poisson Random Variables with Different Time Units

Suppose that W = the number of women who enter a store in an hour, M = the number of men who enter a store in three hours, Z = W + M. Suppose you believe that W ~ Poi(100) (so you have that the ...
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26 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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18 views

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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30 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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127 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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1answer
31 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
35 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 [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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2answers
348 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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45 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
88 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
327 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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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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8 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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55 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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63 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
19 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
42 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
111 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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171 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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608 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
68 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
162 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
47 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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37 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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42 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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109 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
132 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
204 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
64 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
687 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
46 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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193 views

Does the sum of discrete uniforms converge 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
70 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
258 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
466 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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212 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
143 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
178 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
41 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
128 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, $\...