I IID and Dependent RVs: A Closer Look at their Relationship and Parameters

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The discussion focuses on the relationship between independent and dependent random variables (RVs), specifically examining two dependent RVs, X1 and X2, derived from iid standard normal variables ε1 and ε2. The parameters μ and σ are clarified as the mean and standard deviation, respectively, with the standard deviation of X2 being more complex due to its dependence on both ε1 and ε2. The key point of dependency arises because both X1 and X2 are influenced by ε1, leading to their correlation rather than independence. An analogy is made with human arm and leg length, illustrating that while related, the variables do not imply direct causation. Understanding these relationships is crucial for grasping concepts in probability and statistics.
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If ##\epsilon_1,\epsilon_2## are iid ##N(0,1)##, ##X_1=\mu_1+\sigma_1 \epsilon_1## and ##X_2=\mu_2+\rho\epsilon_1+\sigma_2 \epsilon_2## are evidently a pair of dependent RVs that are not identically distributed for most values of the parameters. I have no idea what ##\mu,\sigma,\rho## are. I assume ##\mu## is mean and ##\sigma## is standard deviation? I read this example here.
 
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joshmccraney said:
If ##\epsilon_1,\epsilon_2## are iid ##N(0,1)##, ##X_1=\mu_1+\sigma_1 \epsilon_1## and ##X_2=\mu_2+\rho\epsilon_1+\sigma_2 \epsilon_2## are evidently a pair of dependent RVs that are not identically distributed for most values of the parameters. I have no idea what ##\mu,\sigma,\rho## are. I assume ##\mu## is mean and ##\sigma## is standard deviation? I read this example here.
The example in the link does not say exactly what they are, but we can make the reasonable assumption that he is using the very common notations, where all of the ##\mu##s, ##\epsilon##s, and ##\sigma##s are real number constants EDIT: with ##\epsilon##s, and ##\sigma##s positive. In that case, ##\mu_1## is mean and ##\sigma_1## is standard deviation of ##X_1##. Also, ##\mu_2## is mean of ##X_2##. The standard deviation of ##X_2## is more complicated. The SD of the individual terms is ##\rho## and ##\sigma_2##, but the sum of those has a variance which is the sum of the variances
 
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FactChecker said:
The example in the link does not say exactly what they are, but we can make the reasonable assumption that he is using the very common notations, where all of the ##\mu##s, ##\epsilon##s, and ##\sigma##s are real number constants. In that case, ##\mu_1## is mean and ##\sigma_1## is standard deviation of ##X_1##. Also, ##\mu_2## is mean of ##X_2##. The standard deviation of ##X_2## is more complicated. The SD of the individual terms is ##\rho## and ##\sigma_2##, but the sum of those has a variance which is the sum of the variances
Okay, thanks. So I'm missing the crux: why are these dependent instead of independent? It seems to be because ##\epsilon_1## is a function of ##X_1##, and so ##X_2## implicitly depends on ##X_1##?
 
No. ##\epsilon_1## is not a function of ##X_1##. It is the other way around.
It is not that ##X_2## depends on ##X_1##. It is better to understand that they both depend on ##\epsilon_1##, so their tendencies are related. That makes them correlated and not independent.

(PS. I don't like to say that ##X_1## and ##X_2## are dependent until you are comfortable with what that means in probability. It just means that the tendencies of one give a hint to the tendencies of the other. It does not mean the functional dependency that you are probably used to.)
 
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FactChecker said:
No. ##\epsilon_1## is not a function of ##X_1##. It is the other way around.
It is not that ##X_2## depends on ##X_1##. It is better to understand that they both depend on ##\epsilon_1##, so their tendencies are related. That makes them correlated and not independent.

(PS. I don't like to say that ##X_1## and ##X_2## are dependent until you are comfortable with what that means in probability. It just means that the tendencies of one give a hint to the tendencies of the other. It does not mean the functional dependency that you are probably used to.)
Perfect explanation, thanks so much!
 
Consider human arm length and leg length. Clearly, they are related and not independent, yet there are many other factors involved and one does not cause the other. It is just that their tendencies are related.
 
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For one concrete example, one can see a similar form in Moving Average models in time series analysis

1650297202304.png


https://en.wikipedia.org/wiki/Moving-average_model
 
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