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# Standard Error Of Sum Of Variables

may be challenged and removed. (December 2009) (Learn how and when to remove this template message) In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables, which

## Sum Of Standard Errors

can be quite complex based on the probability distributions of the random variables involved and sum of independent random variables 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

## Sum Of Random Variables Variance

2 Correlated random variables 2.1 Proof 3 References 4 See also Independent random variables If X and Y are independent random variables that are normally distributed (and therefore also jointly so), then their sum is also expected value of sum of random variables 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 , 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 variance of sum of independent random variables 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 Proof using characteristic functions [citation needed] The characteristic function φ X + Y ( t ) = E ⁡ ( e i t ( X + Y ) ) {\displaystyle \varphi _{X+Y}(t)=\operatorname {E} \left(e^{it(X+Y)}\right)} of the sum of two independent random variables X and Y is just the product of the two separate characteristic functions: φ X ( t ) = E ⁡ ( e i t X ) , φ Y ( t ) = E ⁡ ( e i t Y ) {\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),\qquad \varphi _{Y}(t)=\operatorname {E} \le

sample mean. Therefore, we need some results about the properties of sums of random variables. Expected Value For any two random variables $X$ and $Y$, the expected value of the sum of those variables will be equal

## A Certain List Of Zeros And Ones Has Standard Deviation 0.3. The Percentage Of Ones On The List

to the sum of their expected values. $E(X + Y) = E(X) + E(Y)$ The normal distribution proof, for both the discrete and continuous cases, is rather straightforward. The only essential observations are that the order of the summations (or

## Central Limit Theorem

integrals) can be swapped, and that marginal functions occur midway through 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 https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables (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) \, \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 http://www.milefoot.com/math/stat/rv-sums.htm 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\limits_y 2xy P_{XY}(x,y) + \sum\limits_y \sum\limits_x y^2 P_{XY}(x,y) - (E(X))^2 - 2E(X)E(Y) - (E(Y))^2 \\ &= \sum\limits_x x^2 P_X (x) - (E(X))^2 + \sum\limits_y y^2 P_Y(y) - (E(Y))^2 + \sum\limits_x \sum\limits_y 2xy P_{XY}(x,y) - 2E(X)E(Y) \\ &= E(X^2) - (E(X))^2 + E(Y^2) - (E(Y))^2 + 2 (E(XY) - E(X)E(Y) ) \\ &= Var(X) + Var(Y) + 2 Cov(X,Y) \end{align} And now we do the continuous case. \begin{align} &Var(X + Y) \\ &= \int_x \int_y (x+y)^2 f_{XY}(x,y) - (E(X+Y))^2 \\ &= \int_x \int_y x

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 http://stats.stackexchange.com/questions/48133/sum-standard-deviation-vs-standard-error the company Business Learn more about hiring developers or posting ads with us Cross Validated http://stats.stackexchange.com/questions/164505/standard-error-for-sum 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 sum of rise to the top Sum standard deviation vs standard error up vote 3 down vote favorite I'm having difficulty in determining what exactly the difference is between the 2, especially when given an exercise and I have to choose which of the 2 to use. These is how my text book describes them: Sum standard deviation Given is a population with a normally distributed random variable $X$. When you have a sample $n$ of sum of from this population the population is: $X_{sum} = X_1 + X_2 ... + X_n$ with $\mu_{Xsum} = n \times \mu_x$ and $\sigma_{Xsum} = \sqrt{n} \times \sigma_x$. Standard error When you have a normally distributed random variable $X$ with mean $\mu_X$ and standard deviation $\sigma_X$ and sample length $n$, the sample mean $\bar{X}$ is normally distributed with $\mu_{\bar{x}} = \mu_X$ and $\sigma_{\bar{x}} = \dfrac{\sigma_X}{\sqrt{n}}$ These 2 are awefully similair to me to the point I can't at all decide which to use where. Here are the problems where I discovered I couldn't: Problem 1 A filling machine fills bottles of lemonade. The amount is normally distributed with $\mu = 102 \space cl$. $\sigma$ = $1.93\space cl$. Calculate the chance that out of 12 bottles the average volume is $100 \space cl$. The problem itself is easy, however the troublesome part is what to choose for the standard deviation of the sample. Here they use $\dfrac{1.93}{\sqrt{12}}$ which I can live with, until I encountered the second problem. Problem 2 A tea company puts 20 teabags in one package. The weight of a teabag is normally distributed with $\mu = 5.3 \space g$ and $\sigma = 0.5 \space g.$ Calculate the chance that a package weighs less than 100 grams. Here I thought they'd also use $\dfrac{0.5}{\sqrt{20}}$, but instead th

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 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 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 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

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Sum Of Squared Relative Error p be evaluated and compared Root Mean Squared Error RMSE is a popular sum of squares error formula formula to measure the error rate of a regression model residual sum of squares formula However it can only be compared between models whose errors are measured in the same units regression sum of squares Relative Squared Error Unlike RMSE the relative squared error RSE can be compared between models whose errors are measured in the different units Mean Absolute total sum of squares Error The mean absolute error MAE has the same unit as the original

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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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