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Further, two jointly normally distributed random variables are independent if they are uncorrelated, [4] although this does not hold for variables whose marginal distributions are normal and uncorrelated but whose joint distribution is not joint normal (see Normally distributed and uncorrelated does not imply independent).
Students of statistics and probability theory sometimes develop misconceptions about the normal distribution, ideas that may seem plausible but are mathematically untrue. For example, it is sometimes mistakenly thought that two linearly uncorrelated, normally distributed random variables must be statistically independent.
Independent: Each outcome will not affect the other outcome (for from 1 to 10), which means the variables , …, are independent of each other. Identically distributed : Regardless of whether the coin is fair (with a probability of 1/2 for heads) or biased, as long as the same coin is used for each flip, the probability of getting heads remains ...
This example shows that if two random variables are uncorrelated, that does not in general imply that they are independent. However, if two variables are jointly normally distributed (but not if they are merely individually normally distributed), uncorrelatedness does imply independence. [9]
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein. [3]Suppose X and Y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails.
This is a weaker restriction than strict exogeneity, which requires the variable to be uncorrelated with past, present, and future shocks. A common example of a predetermined variable is consumption in models with credit constraints and rational expectations. Here, consumption is predetermined but not strictly exogenous.
A variable omitted from the model may have a relationship with both the dependent variable and one or more of the independent variables (causing omitted-variable bias). [3] An irrelevant variable may be included in the model (although this does not create bias, it involves overfitting and so can lead to poor predictive performance).