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In statistical hypothesis testing, a two-sample test is a test performed on the data of two random samples, each independently obtained from a different given population. The purpose of the test is to determine whether the difference between these two populations is statistically significant .
For example, if two fair six-sided dice are thrown to generate two uniformly distributed integers, and , each in the range from 1 to 6, inclusive, the 36 possible ordered pairs of outcomes (,) constitute a sample space of equally likely events. In this case, the above formula applies, such as calculating the probability of a particular sum of ...
This is not a one-to-one correspondence between {0,1} ∞ and [0,1] however: it is an isomorphism modulo zero, which allows for treating the two probability spaces as two forms of the same probability space. In fact, all non-pathological non-atomic probability spaces are the same in this sense.
That is, the probability function f(x) lies between zero and one for every value of x in the sample space Ω, and the sum of f(x) over all values x in the sample space Ω is equal to 1. An event is defined as any subset E {\displaystyle E\,} of the sample space Ω {\displaystyle \Omega \,} .
A stochastic process is defined as a collection of random variables defined on a common probability space (,,), where is a sample space, is a -algebra, and is a probability measure; and the random variables, indexed by some set , all take values in the same mathematical space , which must be measurable with respect to some -algebra .
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.
A related phenomenon driven by climate change is woody plant encroachment, affecting up to 500 million hectares globally. [218] Climate change has contributed to the expansion of drier climate zones, such as the expansion of deserts in the subtropics. [219] The size and speed of global warming is making abrupt changes in ecosystems more likely ...
In addition to global climate variability and global climate change over time, numerous climatic variations occur contemporaneously across different physical regions. The oceans' absorption of about 90% of excess heat has helped to cause land surface temperatures to grow more rapidly than sea surface temperatures. [ 132 ]