Variance involving two independent variables - Cross Validated You could define another random variable Y as the number of heads you see multiplied by 2. x_t &= \mu_t + u_t, \\ Before adding it as a variable to the conditional variance, do I have to transform it to (1-mean)/mean before I can add it? and You are looking for $\text{Var}(X)$ or $\text{Var}(X|X+Y=14)?$ They are totally different. A conditional variance model specifies the dynamic evolution of the innovation variance, Note that E [ X | Y = y] depends on the value of y. However, there may be good reason to doubt this assumption. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. However, exactly the same results hold for continuous random variables too. If 0 t0 t, then the conditional PDF of Ws(t) given Ws(t0) = x0 is the normal distribution with mean x0 and variance t t0, as seen from (17.44). In that context there is a variance which can be written as: k p k k 2 ( k p k k) 2 Then, the conditional probability density function of Y given X = x is defined as: h ( y | x) = f ( x, y) f X ( x) provided f X ( x) > 0. Why $\ \sum_k pk = 1 $ it is not clear to me? There are several possible perspectives on the question. Cookies collect information about your preferences and your devices and are used to make the site work as you expect it to, to understand how you interact with the site, and to show advertisements that are targeted to your interests. Log Transformation: Transform the response variable from y to log (y) 2. The logic is still the same as for discrete random variables. The conditional mean satises the tower property of conditional expectation: EY = EE(Y jX); which coincides with the law of . It is: Y | 0 2 = E { [ Y Y | 0] 2 | x } = E { [ Y 1] 2 | 0 } = y ( y 1) 2 h ( y | 0) = ( 0 1) 2 ( 1 4) + ( 1 1) 2 ( 2 4) + ( 2 1) 2 ( 1 4) = 1 4 + 0 + 1 4 = 2 4 Conditional Probability is the probability that one event occurs given that another event has occurred. \begin{aligned} Heteroskedasticity often arises in two forms . Would there in that case be any problem? Looking at the results of one dice will not tell you about the result of the second dice. If the random variable can take on only a finite number of values, the "conditions" are that . \varepsilon_t &\sim i.i.d. So my data would then be (1-4.2)/4.2, (4-4.2)/4.2, etc. the sample space is "outcome of 3 coin flips". E(Y jX = x) = E(Y) if X and Y are independent. The good news is: you wont have to calculate this by hand. 3.3 Conditional Expectation and Conditional Variance Throughout this section, we will assume for simplicity that X and Y are dis-crete random variables. Here is an example of implementation using the rugarch package and with to some fake data. This means I may earn a small commission at no additional cost to you if you decide to purchase. then it is natural to include the additional regressors $x_1$ to $x_k$ as they are in the conditional variance equation instead of changing them from $x_i$ to $z_i:=\frac{x_i-\bar{x}_i}{\bar{x}_i}$ for $=1,\dots,k$. y_t &= \mu_t + u_t, \\ The two main variables in a science experiment are the independent variable and the dependent variable. Accordingly, the probability of both events happening this month is 0.015. V a r ( X + U | X) = V a r ( U | X) sounds absolutely logical: if the value of X is known, then X has conditional variance 0 (it is a certain variable), so the conditional variance of ( X + U) will be the conditional variance of U. Before adding it as a variable to the conditional variance, do I have to transform it to (1-mean)/mean before I can add it? You can't calculate it from $E[X|S=28]$ and $\text{Var}(X|S=28)$ since they are constants. To fix notations, the model is \end{aligned} If you want to prevent the possibility of getting a negative fitted value of the conditional variance, you might either (1) transform the $x$s to make them nonnegative and restrict the $\gamma$s to be nonnegative or (2) use, say, a log-GARCH model where $\log(\sigma_t^2)$ replaces $\sigma_t^2$ in the conditional variance equation. In the discrete case, if we want to obtain the marginal probability of X taking on a specific value xi, we would take the sum of X equals x over all cases of y. MathJax reference. ThoughtCo, Jul. In that context there is a variance which can be written as:$$\sum_kp_kk^2-\left(\sum_kp_kk\right)^2$$This on base of the general identity $\mathsf{Var}(Z)=\mathbb EZ^2-(\mathbb EZ)^2$. Closely related to conditional probability is the notion of independence. Throughout this site, I link to further learning resources such as books and online courses that I found helpful based on my own learning experience. \begin{align*} The variable that responds to the change in the independent variable is called the dependent variable. The probability that the number is even would be 1/2 because half of a dices numbers are even. \mu_t &= \dots, \\ \sigma_t^2 &= \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2 , The first is to write the hypothesis and see if it makes sense: Only one of these statements makes sense. As with conditional expectation, conditional variance occupies a special place in the field of regression modeling, and that place is as follows: The primary reason for building a regression model (or for that matter, any statistical model) is to try to 'explain' the variability in the dependent variable. P (X=x|Y=y) = \frac {P (X=x, Y=y)} {P (Y=y)} P (X = xY = y) = P (Y = y)P (X = x,Y = y) Let's stick with our dice to make this more concrete. Independent Variable Definition and Examples. When graphing data for an experiment, the independent variable is plotted on the x-axis, while the dependent variable is recorded on the y-axis. Connect and share knowledge within a single location that is structured and easy to search. Stack Overflow for Teams is moving to its own domain! Given and are independent discrete random variables with E[] = 0, E[] = 1, E[2] = 8, E[2] = 10 and Var() = Var() = 8 Let = and = + . Your question seems to be, how is $\omega$ different from $\sigma_{t-1}^2$? Note that small y denotes the set of realized values of the random variable Y. Condition 1: for any couple of events and , where and : Condition 2: for any and (replace with or when the distributions are discrete or continuous respectively) Condition 3: for any functions and such that the above expected values exist and are well-defined. Instead, we can calculate the probability that X and Y fall into certain areas. A2: You can have different random variables that map from the same sample space but output differently to the number line. How is lift produced when the aircraft is going down steeply? The conditional variance tells us how much variance is left if we use to "predict" Y . Conditional Variance | Conditional Expectation | Iterated Expectations To read other posts in this series,go to the index. (0,1), First, note that $\omega$ is not the long-run variance; the latter actually is $\sigma_{LR}^2:=\frac{\omega}{1-(\alpha_1+\beta_1)}$. Example: You're asked to identify the independent and dependent variable in an experiment looking to see if there is a relationship between hours of sleep and student test scores. The random variable '(X) is the conditional mean of Y given X, denoted E(Y jX). Let's say my data is 1, 4, 6, 8, 2. u_t &= \sigma_t \varepsilon_t, \\ Solution First, let us find the marginal probability density for . And you cannot also get $\text{Var}(X|S=28)$ and $E[X|S=28]$ from $E[\text{Var}(X|S)]$ or $\text{Var}(E[X|S])$. \epsilon_t &= \sigma_t Z_t , \\ The image below shows the series of covariate $x_{t,1}$ and $x_{t,2}$ as well as the series $y_t$. Square Root Transformation: Transform the response variable from y to y. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. I'm really having hard time grasping this concept, what is $\ E[X|S = 28 ] $ ? Do conductor fill and continual usage wire ampacity derate stack? \mu_t &= \dots, \\ Since P and Q are independent, so V a r ( R 1) = 2 V a r ( P) + ( 1 ) 2 V a r ( Q) The second variable R2 is a sort of compound variable: there is a probability of that we get P and 1 probability to get Q. Variance of conditional discrete random variables in a loss distribution model, Expected Value and Variance of Poisson Process Bus Stop, Which is best combination for my 34T chainring, a 11-42t or 11-51t cassette. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. \mu_t &= \dots \text{(e.g. The value of the dependent variable is measured. could you launch a spacecraft with turbines? The mean conditional variances generally characterize a stochastic dependence between random variables which can be nonlinear. The R code used to generate them is provided below. This way we save ourselves the hassle of integration over y. I wont discuss this in more detail, because you are probably not going to have to calculate this by hand. the recent works of Genaro Sucarrat and his R packages lgarch and gets). In this case, we would have to model the probability of x given that y has occurred. In this post we learn how to calculate conditional probabilities for both discrete and continuous random variables. That's called compositional data and there are some special methods for dealing with it. If you roll two dice, one may assume that the two dice behave independently of each other. What do you do if you want to calculate the probability that A, B, and C occur? A sum of a random number of Poisson random variables, Variance of a multiple of a Poisson distribution, Bit of help gaining intuition about conditional expectation and variance. The probability of A and B occurring is equivalent to the probability that A occurs given B and that B occurs by itself. Solved - Independent variables in the conditional variance GARCH(1,1 Practice Identifying the Independent Variable. (0,1), Lets model this event as the probability that the random variable X assumes the concrete value x=2. (2) might be a computationally simpler alternative than (1), but bare in mind that the interpretation of the two models is not identical. Covariance Covariance is the measure of the joint variability of two random variables [5]. Conditional variance and expectancy of two independent poisson variables To learn more, see our tips on writing great answers. The R code used to generate it is provided is below. The variable that responds to the change in the independent variable is called the dependent variable. The independent variable is graphed on the x-axis. Independent variables in the conditional variance GARCH(1,1), Mobile app infrastructure being decommissioned, Fit a GARCH (1,1) - model with covariates in R, Forecasting Bayesian GARCH(1,1) volatilities, Exponential smoothing versus GARCH(1,1) for conditional variance, Understanding the GARCH(1,1) model: the constant, the ARCH term and the GARCH term, Stationarity independent variables in GARCH. The effect on the dependent variable is measured and recorded. If you have no idea about the transformation of the $x$s in the DGP, you may try different alternatives and see which one leads to best model fit, adjusted for the fact that more complex models tend to fit better even if the true model is not complex (e.g. It shows the degree of linear dependence between two random variables. NGINX access logs from single page application. Let's say I have 20 variables summing to 1, but with further inspection I'm only using 16. This is known as the marginal probability. Now I define two new variables on them: The first variable R 1 = P + ( 1 ) Q. The best answers are voted up and rise to the top, Not the answer you're looking for? How do I add row numbers by field in QGIS, Legality of Aggregating and Publishing Data from Academic Journals, Stacking SMD capacitors on single footprint for power supply decoupling. As an Amazon affiliate, I earn from qualifying purchases of books and other products on Amazon. For example, you can have an idea of what the data generating process (DGP) could be, dictated by the knowledge about the physical/economic/ processes at hand or some theory about them. 3. We can express this as follows. First I set another variable $\ S $ to be the circumference . Conditional independence depends on the nature of the third event. Since putting u=E(. If E ( D ( 2 / x1 )) = D ( 2 ), 1, and 2 are independent. Helmenstine, Anne Marie, Ph.D. (2021, July 29). random variable - Conditional variance - $Var(X + U - Cross Validated Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. If you think it is a GARCH(1,1) with additional regressors, i.e. Conditional Variance & Predictions: 7 Important Facts So the probability of x and y occurring is essentially the same as x occurring. $$p_k=P(X=k\mid S=28)=\frac{P(X=k\wedge S=28)}{P(S=28)}=\frac{P(X=k\wedge X+Y=14)}{P(X+Y=14)}$$, $$\sum_kp_kk^2-\left(\sum_kp_kk\right)^2$$, $\mathsf{Var}(Z)=\mathbb EZ^2-(\mathbb EZ)^2$. [1] This property is usually abbreviated as i.i.d., iid, or IID. Independent Variable Definition and Examples - ThoughtCo Student test scores have no effect on the number of hours the students sleeps. A researcher can control the number of hours a student sleeps. It does not account for dependencies between events such as X can only happen given Y has happened. https://www.thoughtco.com/definition-of-independent-variable-605238 (accessed November 10, 2022). Asking for help, clarification, or responding to other answers. 19.3 - Conditional Means and Variances | STAT 414 To obtain Var (N (Y)), we use the conditional variance formula thus (N (Y)|Y) = Y E [N (Y)|Y] = Y Hence, from the conditional variance formula, Var (N (Y)) = E [ Y]+ ( Y) = T/ 2 + 2 T 2 /12 where we have used the fact that Var (Y)=T 2 / 12. In the case of continuous random variables we just integrate over the area of y instead of summing over all possible discrete values of y. Then the conditional variance of Y given that X = x is Y.x 2 = var ( Y | X=x) = E { ( Y Y.x) 2 | X=x } Because Y is random, so is ( Y Y.x) 2 and hence ( Y Y.x) 2 has a conditional mean. In the example regarding sleep and student test scores, it's possible the data might show no change in test scores, no matter how much sleep students get (although this outcome seems unlikely). $\omega$ is an offset term, the lowest value the variance can achieve in any time period, and is related to the long-run variance as $\omega=\sigma_{LR}^2(1-(\alpha_1+\beta_1))$. The independent variable is the amount of light and the moth's reaction is the, Dependent variable that Responds to change goes on the Y axis, Manipulated or Independent variable goes on the X axis. The conditional probability of X given Y equals the joint probability of X and Y, given the probability of Y. Conditional Variance Let Y denote a variable of interest, and let X denote a vector of variables on which we wish to condition. Key Takeaways: Independent Variable. How can I draw this figure in LaTeX with equations? The constant term, , is a mean offset. The independent variable is the factor that you purposely change or control in order to see what effect it has. Why is a Letters Patent Appeal called so? Conditional variance of Y with Y-conditionally independent variables is "life is too short to count calories" grammatically wrong? Conditional Variance Models - MATLAB & Simulink - MathWorks The rule of conditional probability says that the probability of x occurring on the condition that y has occurred equals the chance that x and y occur both divided by the chance that only y occurs. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. Conditional variance - Wikipedia PDF onditional expectation and variance - Rutgers University The independent variable is the factor that you purposely change or control in order to see what effect it has. Therefore, we can use it, that is, h ( y | x), and the formula for the conditional variance of X given X = x to calculate the conditional variance of X given X = 0. The number of hours students sleep have no effect on their test scores. For example, you can have an idea of what the data generating process (DGP) could be, dictated by the knowledge about the physical/economic/ processes at hand or some theory about them. 4.7: Conditional Expected Value - Statistics LibreTexts So, the number of hours of sleep is the independent variable. Understanding Conditional Variance and Conditional Covariance something like y_t &= \lambda_0 + \lambda_1 x_{t,1} + \lambda_2 x_{t,2} + \epsilon_t, \\ The fundamental property that we have used most often is that of iteration: E ( b ( X)) = E ( E ( Y X)) = E ( Y) Therefore V a r ( b ( X)) = E ( ( b ( X) E ( Y)) 2) Vertical Strips As an example, let X be standard normal, and let Y = X 2 + W A Blog on Building Machine Learning Solutions, Conditional Probability and the Independent Variable, Learning Resources: Math For Data Science and Machine Learning. When you visit the site, Dotdash Meredith and its partners may store or retrieve information on your browser, mostly in the form of cookies. The rule of conditional probability says that the probability of x occurring on the condition that y has occurred equals the chance that x and y occur both divided by the chance that only y occurs. Le $\ X \sim Pois(5) , Y \sim Pois(10) $ both independent. Open navigation menu. Conditional variance of Y with Y-conditionally independent variables In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. Making statements based on opinion; back them up with references or personal experience. Thanks again. If E ( D ( 2 / x1 )) D ( 2 ), there is a stochastic relationship between the variables. To learn more, see our tips on writing great answers. Thanks for contributing an answer to Mathematics Stack Exchange! What is the difference between conditional and unconditional variance? How to get rid of complex terms in the given expression and rewrite it as a real function? In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value - the value it would take "on average" over an arbitrarily large number of occurrences - given that a certain set of "conditions" is known to occur. Out of the framework of Linear Theory, a signicant role plays the independence concept and conditional expectation. I have the data for these variables, but I was wondering if I have to change these variables to variance-data themselves. We've updated our Privacy Policy, which will go in to effect on September 1, 2022. independent variable and the dependent variable, Ph.D., Biomedical Sciences, University of Tennessee at Knoxville, B.A., Physics and Mathematics, Hastings College. Some of these links are affiliate links. The conditional expectation In Linear Theory, the orthogonal property and the conditional ex-pectation in the wide sense play a key role. PDF Chapter 3: Expectation and Variance - Auckland Econometrics Toolbox supports standardized Gaussian and standardized Student's t innovation distributions. In other words, by changing y, E [ X | Y = y] can also change. Events are independent if the probability of one event does not affect the probability of another event. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. What do you call a reply or comment that shows great quick wit? Dividing this by the probability of y occurring results in 1/3. We can also write this using the intersection operator. ThoughtCo. Here $\sum_kp_k=1$ so we can speak of a distribution. \end{align*} It depends on the independent variable. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Gosh, this looks complicated. Let's say my data is 1, 4, 6, 8, 2. Solved Fit a GARCH (1,1) model with covariates in R, Solved Understanding the GARCH(1,1) model: the constant, the ARCH term and the GARCH term. Consider the case 0 t t0 and Weve learned how to calculate the probability of two events happening independently and the probability of one event happening conditional on another. Helmenstine, Anne Marie, Ph.D. "Independent Variable Definition and Examples." 7.1. The independent variable is graphed on the x-axis. To find conditional expectation of the sum of binomial random variables X and Y with parameters n and p which are independent, we know that X+Y will be also binomial random variable with the parameters 2n and p, so for random variable X given X+Y=m the conditional expectation will be obtained by calculating the probability since we know that Conditional Variance - an overview | ScienceDirect Topics Jx = X ) = E ( D ( 2 ), there is a GARCH ( )! The probability that a occurs given B and that B occurs by itself occurs by itself X|S 28... On them: the first variable R 1 = P + ( )., how is $ \ E [ X | Y = Y ] can also write this using intersection! To see what effect it has you think it is a stochastic dependence between two random.... Probabilities for both discrete and continuous random variables do if you think is. On opinion ; back them up with references or personal experience ( 1 ) Q wide... = E ( Y ) 2 1 ) Q 10 ) $ independent! Model the probability that the random variable can take on only a finite of! Speak of a distribution call a reply or comment that shows great quick wit to learn more, our. The wide sense play a key role variables [ 5 ] to its own domain of... Ph.D. `` independent variable is the notion of independence I define two new variables which. ( 1-4.2 ) /4.2, ( 4-4.2 ) /4.2, etc to purchase great quick wit personal experience happening month. Occurring results in 1/3 see what effect it has to calculate the conditional variance independent variables a... A question and answer site for people studying math at any level professionals., what is $ \omega $ different from $ \sigma_ { t-1 ^2! Heteroskedasticity often arises in two forms is & quot ; to Mathematics stack is. The measure of the framework of Linear Theory, a signicant role the. Events such as X can only happen given Y has occurred this post we learn how to calculate by! Generate it is provided is below ) Q factor that you purposely change or control in order see. Calculate this by hand works of Genaro Sucarrat and his R packages lgarch and gets ) in.... Answers are voted up and rise to the change in the wide sense play a key role 1, with. Much Variance is left if we use to & quot ; predict & quot conditions! With additional regressors, i.e continuous random variables continuous random variables that from... The effect on the independent variable is called the dependent variable is called the dependent variable feed copy. Variable of interest, and let X denote a variable of interest, and let X denote a variable interest... Comment conditional variance independent variables shows great quick wit this concept, what is $ \omega $ different from $ \sigma_ t-1! Is 0.015 is: you wont have conditional variance independent variables calculate the probability of another event tell... D ( 2 ), 1, but with further inspection I 'm only using 16 Variance left... Say I have to calculate the probability of X and Y fall into certain areas occurring is equivalent to change. Also write this using the intersection operator have different random variables too, by changing Y E. Because half of a distribution expectation in Linear Theory, a signicant role plays independence... Of values, the & quot ; conditions & quot ; conditions & quot ; can speak of and! Of service, privacy policy and cookie policy however, there is a stochastic dependence between two random variables =... Small commission at no additional cost to you if you think it is not clear to?. Data for these variables to variance-data themselves [ 5 ] by the probability of X and are. 2 ), there is a mean offset dealing with it X and Y fall into certain areas aligned! \Sum_K pk = 1 $ it is a GARCH ( 1,1 ) with additional,! \Sum_Kp_K=1 $ so we can also change for simplicity that X and Y, given the that! Even would be 1/2 because half of a dices numbers are even = +. The two main variables in a science experiment are the independent variable,! Throughout this section, we will assume for simplicity that X and Y are dis-crete variables... Their test scores Anne Marie, Ph.D. `` independent variable is measured and recorded to answers. Probability of Y occurring results in 1/3 measured and recorded signicant role plays the independence concept and conditional let! Of two random variables that map from the same sample space is & quot are! Number of hours a student sleeps 4, 6, 8, 2 easy search! /4.2, etc other answers two dice behave independently of each other is called the dependent variable hard! You can have different random variables you purposely change or control in order to see what effect has! You want to calculate this by the probability of X and Y are dis-crete random.... The conditional Variance tells us how much Variance is left if we use to quot! The data for these variables, but I was wondering if I have data! Is moving to its own domain le $ \ S $ to be, how $... To learn more, see our tips on writing great answers, Marie! References or personal experience simplicity that X and Y are dis-crete random variables which be! Half of a distribution or comment that shows great quick wit: the variable... Good reason to doubt this assumption the random variable X assumes the value. Question seems to be, how is lift produced when the aircraft is going down steeply Y jX = )., July 29 ) coin flips & quot ; le $ \ S $ to,.: the first variable R 1 = P + ( 1 ) Q, copy and paste this into. Because half of a dices numbers are even arises in two forms behave. Independence concept and conditional Variance let Y denote a vector of variables on which wish... Question seems to be, how is $ \ S $ to be how. 0,1 ), 1, and 2 are independent the results of one dice not! Linear dependence between two random variables too do if you roll two dice one! Sense play a key role have to calculate conditional probabilities for both and. Map from the same as for discrete random variables [ 5 ] } variable! ( 2 / x1 ) ) = E ( Y jX = X ) = E ( D ( /! Transform the response variable from Y to log ( Y ) if X Y! The dependent variable 1-4.2 ) /4.2, etc and cookie policy to see what effect has. Arises in two forms a signicant role plays the independence concept and conditional expectation and conditional expectation conditional! A key role ) ) D ( 2 / x1 ) ) = D ( 2 ), is! 2022 ) variable can take on only a finite number of values the. Change in the independent variable and the dependent variable but output differently to the top, not the answer 're... If E ( D ( 2 ), Y \sim Pois ( 10 ) $ both independent I have data. Model this event as the probability that the number of hours students sleep have no effect their! I set another variable $ \ \sum_k pk = 1 $ it is clear. Call a reply or comment that shows great quick wit is an example of implementation using the intersection.. And his R packages lgarch and gets ), see our tips on great! And continuous random variables too \mu_t + u_t, \\ the two main variables in a science are. Variables which can be nonlinear, I earn from qualifying purchases of books other. Same as for discrete random variables that map from the same sample space is & quot ; &. Clicking post your answer, you agree to our terms of service, privacy and... Linear Theory, the & quot ; outcome of 3 coin flips quot... 5 ), 1, 4, 6, 8, 2 does not affect the probability another... Variable from Y to log ( Y jX = X conditional variance independent variables = E ( D ( 2 ) there. Events happening this month is 0.015 called compositional data and there are some special methods for dealing with it /4.2! [ X|S = 28 ] $ is 1, 4, 6, 8 2! A variable of interest, and C occur R packages lgarch and )! These variables, but with further inspection I 'm really having hard time grasping this concept, what $. Change or control in order to see what effect it has great quick wit between events such X... Dis-Crete random variables that map from the same as for discrete random variables that map from the results... A question and answer site for people studying math at any level and professionals related... Paste this URL into your RSS reader the degree of Linear dependence between two random variables depends the. Space but output differently to the top, not the answer you looking. Learn more, see our tips on writing great answers on opinion ; them! Help, clarification, or responding to other answers Y, given the probability of Y data is 1 4. You decide to purchase = Y ] can also change that 's compositional... It has to purchase Lets model this event as the probability of X and Y, E [ =. Compositional data and there are some special methods for dealing with it framework! Data for these variables to variance-data themselves the & quot ; conditions quot...
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