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Standard Error Of A Sum Of Random Variables

may be challenged and removed. (December 2009) (Learn how and when to remove this template message) In probability theory, calculation of the sum variance of the sum of two random variables of normally distributed random variables is an instance of the arithmetic sum of independent random variables of random variables, which can be quite complex based on the probability distributions of the random variables involved sum of standard errors and their relationships. Contents 1 Independent random variables 1.1 Proofs 1.1.1 Proof using characteristic functions 1.1.2 Proof using convolutions 1.1.3 Geometric proof 2 Correlated random variables 2.1 Proof expected value of sum of random variables 3 References 4 See also Independent random variables[edit] If X and Y are independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed. i.e., if X ∼ N ( μ X , σ X 2 ) {\displaystyle X\sim N(\mu _{X},\sigma _{X}^{2})} Y ∼ N ( μ Y ,

Variance Of Sum Of Independent Random Variables

σ Y 2 ) {\displaystyle Y\sim N(\mu _{Y},\sigma _{Y}^{2})} Z = X + Y , {\displaystyle Z=X+Y,} then Z ∼ N ( μ X + μ Y , σ X 2 + σ Y 2 ) . {\displaystyle Z\sim N(\mu _{X}+\mu _{Y},\sigma _{X}^{2}+\sigma _{Y}^{2}).} This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). Note that the result that the sum is normally distributed requires the assumption of independence, not just uncorrelatedness; two separately (not jointly) normally distributed random variables can be uncorrelated without being independent, in which case their sum can be non-normally distributed (see Normally distributed and uncorrelated does not imply independent#A symmetric example). The result about the mean holds in all cases, while the result for the variance requires uncorrelatedness, but not independence. Proofs[edit] Proof usin

sample mean. Therefore, we need some results about the properties of sums of random variables. Expected Value For any two random variables $X$

Normal Distribution

and $Y$, the expected value of the sum of those variables will central limit theorem be equal to the sum of their expected values. $E(X + Y) = E(X) + E(Y)$ The proof, for convolution both the discrete and continuous cases, is rather straightforward. The only essential observations are that the order of the summations (or integrals) can be swapped, and that marginal functions occur midway through https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables the proof. \begin{align} E(X + Y) &= \sum\limits_x \sum\limits_y (x+y) P_{XY}(x,y) \\ &= \sum\limits_x \sum\limits_y x P_{XY}(x,y) + \sum\limits_y \sum\limits_x y P_{XY}(x,y) \\ &= \sum\limits_x x P_X (x) + y P_Y (y) \\ &= E(X) + E(Y) \end{align} \begin{align} E(X + Y) &= \int_x \int_y (x+y) f_{XY}(x,y) \, \mathrm{d}x \\ &= \int_x \int_y x f_{XY}(x,y) \, \mathrm{d}y \mathrm{d}x + \int_y \int_x y f_{XY}(x,y) http://www.milefoot.com/math/stat/rv-sums.htm \, \mathrm{d}x \mathrm{d}y \\ &= \int_x x f_X (x) \, \mathrm{d}x + \int_y y f_Y (y) \, \mathrm{d}y \\ &= E(X) + E(Y) \end{align} As an example, suppose we have a random variable $Z$ which is the sum of two other random variables $X$ and $Y$. The first has mean $E(X) = 17$ and the second has mean $E(Y) = 24$. Since $Z = X + Y$, then the mean of $Z$ is $E(Z) = 24+17 = 41$. The actual shape of each distribution is irrelevant. Variance For any two random variables $X$ and $Y$, the variance of the sum of those variables is equal to the sum of the variances plus twice the covariance. $Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X,Y)$ The proof of this statement is similar to the proof of the expected value of a sum of random variables, but since variance is involved, there are a few more details that need attention. \begin{align} &Var(X + Y) \\ &= \sum\limits_x \sum\limits_y (x+y)^2 P_{XY}(x,y) - (E(X+Y))^2 \\ &= \sum\limits_x \sum\limits_y x^2 P_{XY}(x,y) + \sum\limits_x \sum\li

Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn http://stats.stackexchange.com/questions/164505/standard-error-for-sum more about hiring developers or posting ads with us Cross Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Standard Error for Sum sum of up vote 1 down vote favorite I am designing an algorithm for a stratified sampling on a population and then I want to find out what is the error bound for 95% confidence interval, for different sample statistic such as sum of the samples. For this purpose, I need to find Standard Error for sum. I cannot find any information on Standard error other than for mean and proportion. My purpose is SE for sum.. Any information on sum of random the formula for SE for SUM, or any links which gives an idea is highly appreciated ! (I have heard SE calculation differs for different sampling techniques, any info about that would be great too). sampling standard-deviation standard-error stratification share|improve this question edited Aug 3 '15 at 17:21 whuber♦ 146k18285547 asked Aug 3 '15 at 15:02 user3218207 61 2 The mean and the sum are usually related by a constant multiple--typically, the mean is $1/n$ times the sum. Since it is obvious that their standard errors will be related by the same multiple, people rarely stop to give explicit formulas for both sums and means: one formula is good enough. –whuber♦ Aug 3 '15 at 17:24 1 Describe your sampling i more detail, show equations –Aksakal Aug 3 '15 at 17:25 add a comment| 1 Answer 1 active oldest votes up vote 0 down vote If you assume that $X_1, ..., X_n$ is a sample from a Gaussian distribution $N(\mu, \sigma^2)$, then this theorem stipulates that a linear combination of the data, $\sum_i c_i X_i$, has distribution $$Y := \sum_{i=1}^{n} c_i X_i \sim N\left( \sum_{i} c_i \mu, \ \sum_{i} c_{i}^2\sigma^2 \right)$$ In your example, $c_i =1 \ \forall i$, so defining $Y$ as the sum, $$Y \sim N\left(n\mu, n\sigma^2 \right)$$ which means that the standard error (i.e. the standard deviation of the sampling distribution of $Y$) is $\sqrt{n}\s

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Sum Of Squared Prediction Error In R p PRESS differs from p Press Statistic In R p the sum of squares of the residual error in that p Press Statistic Interpretation p each fitted value Y hati for PRESS is obtained from the remaining n observations then p Residual Sum Of Squares Example p using the fitted regression function to obtain the predicted value for the ith observation Use PRESS to assess your model's predictive ability Usually the smaller the PRESS value the better the model's predictive sum of squared errors formula ability PRESS is used to calculate the predicted

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Sum Of Squares Of Error Formula p be equal to The formula for SSE is Where n is the number of observations xi is the value of the ith observation and is the mean of all the observations This can also be rearranged to be written as seen in J H p Sum Of Squares Error Calculator p Ward's paper At each stage of cluster analysis the total SSE is minimized with SSEtotal SSE sum of squared errors example SSE SSE SSE SSEn At the initial stage when each case is its own cluster this of course will be You

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Sum Of Squares Due To Error Sse p may be challenged and removed April Learn how and when to remove this template message In statistics the residual sum of squares RSS also known as the sum of squared sum of squares error calculator residuals SSR or the sum of squared errors of prediction SSE is the p Sum Of Squared Errors Example p sum of the squares of residuals deviations predicted from actual empirical values of data It is a measure of sum of squared errors excel the discrepancy between the data and an estimation model A small RSS indicates

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Sum Of Squared Error Criterion p deviation MSD of an estimator of a procedure for estimating an unobserved quantity measures the average of the squares of the errors or deviations that is the difference between the estimator and what is estimated MSE is a risk function corresponding to the expected value of the squared error loss or quadratic loss The difference occurs because of randomness or because the estimator doesn't account for information that could produce a more accurate estimate The MSE is a measure of the quality of an estimator it is always non-negative and values closer to zero

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Sum Of Square Error Formula p be equal to The formula for SSE is Where n is the number of observations xi is the value of the ith observation and is the mean of all the observations This can also be rearranged to be written p Sum Of Squares Error Calculator p as seen in J H Ward's paper At each stage of cluster analysis the total SSE is sum of squared errors example minimized with SSEtotal SSE SSE SSE SSE SSEn At the initial stage when each case is its own cluster this p Sum Of Squared Errors Excel

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Sum Of Squares Due To Error p may be challenged and removed April Learn how and when to remove this template message In statistics the residual sum of squares RSS also known as the sum of squared residuals SSR or the sum of squares error calculator sum of squared errors of prediction SSE is the sum of the squares sum of squared errors example of residuals deviations predicted from actual empirical values of data It is a measure of the discrepancy between the data total sum of squares and an estimation model A small RSS indicates a tight fit of

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Sum Of Squares Error Calculator Online p Tables Constants Calendars Theorems Mean Squared Error Sum of Squared Error sum of squares error formula Calculator Calculator Formula Download sst calculator Script Calculate the mean squared error and sum of squared error p How To Calculate Mean Square Error Example p using this simple online calculator Enter the population values to know the squared errors Mean Square Error Sum of p Sst Ssr Sse Calculator p Squared Error Calculation Enter the Population Values Separated by comma Ex Number of Population n Mean mu Sum of Squared Error SSE Mean Squared Error MSE

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Total Sum Of Squared Error p may be challenged and removed April Learn how and when to remove this template message In statistics the residual sum of squares RSS also known as the sum of squared residuals regression sum of squares SSR or the sum of squared errors of prediction SSE is the sum sum of squares residual of the squares of residuals deviations predicted from actual empirical values of data It is a measure of the error sum of squares discrepancy between the data and an estimation model A small RSS indicates a tight fit of the model to

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Unexplained Variation Or Error Sum Of Squares Sse p may be challenged and removed April Learn how and when to remove this template message In statistics the residual sum of squares RSS also known as the sum of squared residuals SSR or the sum of squared how to calculate ssr errors of prediction SSE is the sum of the squares of residuals deviations sst ssr sse predicted from actual empirical values of data It is a measure of the discrepancy between the data and an estimation model p Sst Statistics p A small RSS indicates a tight fit of the

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What Does Ss Error Mean p may be challenged and removed April Learn how and when to remove this template message In statistics the residual sum of squares RSS also known as the sum of squared residuals SSR p Error Sum Of Squares p or the sum of squared errors of prediction SSE is the sum of sum of squares error calculator the squares of residuals deviations predicted from actual empirical values of data It is a measure of the discrepancy between total sum of squares the data and an estimation model A small RSS indicates a tight fit of

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What Does Sum Of Squares Error Mean p is uncorrected sum of squares What is sum of squares The sum of squares represents a measure of variation or deviation from the mean It is calculated as a summation of the squares of the regression sum of squares differences from the mean The calculation of the total sum of squares considers p Total Sum Of Squares p both the sum of squares from the factors and from randomness or error Sum of squares in ANOVA In p Sum Of Squares Error Calculator p analysis of variance ANOVA the total sum of