Search results
Results From The WOW.Com Content Network
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.
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).
As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables. If the variables are independent, Pearson's correlation coefficient is 0. However, because the correlation coefficient detects only linear dependencies between two ...
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.
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]
In this case, if X and Y are uncorrelated, i.e., their covariance cov(X, Y) is zero, then they are 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 ...
See Chapter 17 of Testing Statistical Hypotheses for example. [9] The distribution is particularly useful in high-dimensional statistics. [10] The Rademacher distribution can be used to show that normally distributed and uncorrelated does not imply independent.