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Pairwise independent random variables with finite variance are uncorrelated. A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF) , (,) satisfies , (,) = (),
The pair distribution function describes the distribution of distances between pairs of particles contained within a given volume. [1] Mathematically, if a and b are two particles, the pair distribution function of b with respect to a, denoted by () is the probability of finding the particle b at distance from a, with a taken as the origin of coordinates.
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.
Partition the sequence into non-overlapping pairs: if the two elements of the pair are equal (00 or 11), discard it; if the two elements of the pair are unequal (01 or 10), keep the first. This yields a sequence of Bernoulli trials with p = 1 / 2 , {\displaystyle p=1/2,} as, by exchangeability, the odds of a given pair being 01 or 10 are equal.
where S is the standard deviation of D, Φ is the standard normal cumulative distribution function, and δ = EY 2 − EY 1 is the true effect of the treatment. The constant 1.645 is the 95th percentile of the standard normal distribution, which defines the rejection region of the test. By a similar calculation, the power of the paired Z-test is
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
Given a large enough pool of variables for the same time period, it is possible to find a pair of graphs that show a spurious correlation. In statistics , the multiple comparisons , multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously [ 1 ] or estimates a subset of parameters selected ...
where is the Kullback–Leibler divergence, and is the outer product distribution which assigns probability () to each (,).. Notice, as per property of the Kullback–Leibler divergence, that (;) is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when and are independent (and hence observing tells you nothing about ).