Dependently Identically Distributed/Correlated Identically Distributed
 Python Automation and Machine Learning for ICs   An Online Book  

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================================================================================= When we drop the "Independent" condition from Independent and Identically Distributed, we left with a scenario where the variables or data points, e.g. a set of variables , are only "Identically Distributed." . In this case, the variables still share the same probability distribution, meaning they have the same range of possible values and follow the same probability density function (PDF) or probability mass function (PMF). However, there is no requirement for them to be independent of each other. In mathematical terms, the correlation between can be expressed as Corr( for all and in the set. The value of indicates the strength and direction of the linear relationship between the variables. If is positive, it suggests a positive linear correlation, and if is negative, it suggests a negative linear correlation. If is zero, it indicates no linear correlation.This scenario is sometimes referred to as "Dependently Identically Distributed" or "Correlated Identically Distributed." It implies that while the variables have the same distribution, the occurrence or value of one variable may be influenced by or dependent on the others in the set. The correlation or dependence structure among the variables is not constrained, and they may exhibit some form of interdependence. The variance of the sample mean can be calculated using the formula:  [3741a] where, n is the sample size. is the variance of a single variable. Corr( is the covariance between any two variables X_{i} and X_{j} in the set.If the variables are independent, then the covariance term becomes zero, that is, Corr( = 0 for all i ≠ , then the formula simplifies to: [3741b] In the case of dependent variables, the covariance term introduces additional variability in the sample mean, reflecting the correlation among the variables. For the variance of the sample mean ( ) when dealing with a scenario where the variables are correlated, we have, [3741c] where,
In the case when the variables are correlated, we need to take into account the covariance between each pair of variables. Therefore, the variance of the sample mean is given by: [3741ab] In the expressions in Equations 3741a and 3741c,  [3741d]  [3741e] If the variables are not correlated (independent), then Equation 3741ab becomes,  [3741f] It then is Independent and Identically Distributed (i.i.d). Figure 3741 shows the comparison between IID (Independent and Identically Distributed) and DID (Dependent Identically Distributed). The blue histograms represent the distributions of the variables when the variables are Independent and Identically Distributed. The orange histograms represent the distributions of the same variables, but now taking into account the correlation with other variables. The presence of the orange histograms indicate how the variable's distributions are influenced by the correlations with other variables. (a) (b)
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