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  2. Normal distribution - Wikipedia

    en.wikipedia.org/wiki/Normal_distribution

    The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...

  3. Word problem (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Word_problem_(mathematics)

    The most direct solution to a word problem takes the form of a normal form theorem and algorithm which maps every element in an equivalence class of expressions to a single encoding known as the normal form - the word problem is then solved by comparing these normal forms via syntactic equality. [1]

  4. Cochran's theorem - Wikipedia

    en.wikipedia.org/wiki/Cochran's_theorem

    where F 1,n − 1 is the F-distribution with 1 and n − 1 degrees of freedom (see also Student's t-distribution). The final step here is effectively the definition of a random variable having the F-distribution.

  5. Zipf's law - Wikipedia

    en.wikipedia.org/wiki/Zipf's_law

    Empirically, a data set can be tested to see whether Zipf's law applies by checking the goodness of fit of an empirical distribution to the hypothesized power law distribution with a Kolmogorov–Smirnov test, and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential ...

  6. List of probability distributions - Wikipedia

    en.wikipedia.org/wiki/List_of_probability...

    The generalized normal distribution; The geometric stable distribution; The Gumbel distribution; The Holtsmark distribution, an example of a distribution that has a finite expected value but infinite variance. The hyperbolic distribution; The hyperbolic secant distribution; The Johnson SU distribution; The Landau distribution; The Laplace ...

  7. Independent and identically distributed random variables

    en.wikipedia.org/wiki/Independent_and...

    The i.i.d. assumption is also used in the central limit theorem, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution. [4] The i.i.d. assumption frequently arises in the context of sequences of random variables. Then, "independent and identically ...

  8. Student's t-test - Wikipedia

    en.wikipedia.org/wiki/Student's_t-test

    X follows a normal distribution with mean μ and variance σ 2 /n. s 2 (n − 1)/σ 2 follows a χ 2 distribution with n − 1 degrees of freedom. This assumption is met when the observations used for estimating s 2 come from a normal distribution (and i.i.d. for each group). Z and s are independent.

  9. Standard normal table - Wikipedia

    en.wikipedia.org/wiki/Standard_normal_table

    Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. The standard normal distribution, represented by Z, is the normal distribution having a mean of 0 and a standard deviation of 1.