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Diagram showing the cumulative distribution function for the normal distribution with mean (μ) 0 and variance (σ 2) 1. These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score z corresponds numerically to (1 − (1 − Φ μ,σ 2 (z)) · 2).
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
Law of the unconscious statistician: The expected value of a measurable function of , (), given that has a probability density function (), is given by the inner product of and : [34] [()] = (). This formula also holds in multidimensional case, when g {\displaystyle g} is a function of several random variables, and f {\displaystyle f} is ...
In statistics, an empirical distribution function (a.k.a. an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical measure of a sample. [1] This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. Its value at any specified value of the ...
Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value. Frequency distribution: a table that displays the frequency of various outcomes in a sample.
The moment generating function of a real random variable is the expected value of , as a function of the real parameter . For a normal distribution with density f {\displaystyle f} , mean μ {\displaystyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the moment generating function exists and is equal to
It is a transformation of the four-parameter beta distribution with an additional assumption that its expected value is μ = a + 4 b + c 6 . {\displaystyle \mu ={\frac {a+4b+c}{6}}.} The mean of the distribution is therefore defined as the weighted average of the minimum, most likely and maximum values that the variable may take, with four ...
Frequency analysis [2] is the analysis of how often, or how frequently, an observed phenomenon occurs in a certain range. Frequency analysis applies to a record of length N of observed data X 1, X 2, X 3. . . X N on a variable phenomenon X. The record may be time-dependent (e.g. rainfall measured in one spot) or space-dependent (e.g. crop ...