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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.
However, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. Two random variables that are normally distributed may fail to be jointly normally distributed, i.e., the vector whose components they are may fail to have a multivariate normal distribution.
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.
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.
Small business owners should not forget about a rule — currently in legal limbo — that would require them to register with an agency called the Financial Crimes Enforcement Network, or FinCEN ...
In this particular example, if we do not know beforehand that the two types of intelligence are uncorrelated, then we cannot interpret the two factors as the two different types of intelligence. Even if they are uncorrelated, we cannot tell which factor corresponds to verbal intelligence and which corresponds to mathematical intelligence ...
Showers are raining down over the middle of the country, bringing thunderstorms and locally heavy rainfall that is moving across east Texas, southeast Oklahoma and Arkansas.