“cumulative distribution function”的概念、定义、翻译、参考文献-科学参考

cumulative distribution function

Mathematics

For a random variable X, the cumulative distribution function (or cdf) is the function F_X defined by F_X(x) = Pr(X ≤ x). Thus, for a discrete random variable,

$F X (x) = \sum_{x_{i} \leq x} p (x_{i}),$

where p_X is the probability mass function, and, for a continuous random variable, with a probability density function f_X, then $F_{X} (x) = \int_{- \infty}^{x} f_{X} (t) d t .$

Statistics

The function F, for a random variable X, defined for all real values of x by $cumulative distribution function$ Clearly, F(−∞)=0, and F(∞)=1, where F(−∞) and F(∞) are the limits of F(x) as x tends to −∞ and ∞, respectively. This function is a non-decreasing function such that if x₂>x₁ then F(x₂)≥F(x₁). If X is a continuous random variable then F is a continuous function, and conversely. If X has probability density function f then $cumulative distribution function$ and f(x)=F′(x), where F′(x) denotes the derivative of F(x).
A useful property of F is that, for any value of x, there is a corresponding value u, 0≤u≤1, such that $cumulative distribution function$
In the case where F is a continuous and increasing function for a≤x≤b, the random variable U defined by U=F(X) has a continuous uniform distribution on the interval 0≤u≤1, and, for a given value of U, the corresponding value of X is given by F⁻¹(U). See simulation.
In the case of a discrete random variable, the distribution function is a step function, in which the step at x_j is P(X=x_j), and F(x) → F(x_j) as x → x_j from above, but F(x) → F(x_j−1) as x → x_j from below.
Cumulative distribution function. In each example the probability density function f defined in the interval (a, b) is shown on the left, with the corresponding cumulative distribution function F on the right. The scale from 0 to 1 refers to F.

Computer

See probability distributions.

Economics

The cdf of a random variable X at point x is the probability that X takes values at or below this point. Hence, F(x) = P[X ≤ x]. The following properties must hold for any cdf: it is a non-decreasing right-continuous function; it tends to 0 at minus infinity and tends to 1 at plus infinity.

单词	cumulative distribution function
释义	cumulative distribution function Mathematics For a random variable X, the cumulative distribution function (or cdf) is the function F_X defined by F_X(x) = Pr(X ≤ x). Thus, for a discrete random variable, $F X (x) = \sum_{x_{i} \leq x} p (x_{i}),$ where p_X is the probability mass function, and, for a continuous random variable, with a probability density function f_X, then $F_{X} (x) = \int_{- \infty}^{x} f_{X} (t) d t .$ Statistics The function F, for a random variable X, defined for all real values of x by $cumulative distribution function$ Clearly, F(−∞)=0, and F(∞)=1, where F(−∞) and F(∞) are the limits of F(x) as x tends to −∞ and ∞, respectively. This function is a non-decreasing function such that if x₂>x₁ then F(x₂)≥F(x₁). If X is a continuous random variable then F is a continuous function, and conversely. If X has probability density function f then $cumulative distribution function$ and f(x)=F′(x), where F′(x) denotes the derivative of F(x). A useful property of F is that, for any value of x, there is a corresponding value u, 0≤u≤1, such that $cumulative distribution function$ In the case where F is a continuous and increasing function for a≤x≤b, the random variable U defined by U=F(X) has a continuous uniform distribution on the interval 0≤u≤1, and, for a given value of U, the corresponding value of X is given by F⁻¹(U). See simulation. In the case of a discrete random variable, the distribution function is a step function, in which the step at x_j is P(X=x_j), and F(x) → F(x_j) as x → x_j from above, but F(x) → F(x_j−1) as x → x_j from below. Cumulative distribution function. In each example the probability density function f defined in the interval (a, b) is shown on the left, with the corresponding cumulative distribution function F on the right. The scale from 0 to 1 refers to F. Computer See probability distributions. Economics The cdf of a random variable X at point x is the probability that X takes values at or below this point. Hence, F(x) = P[X ≤ x]. The following properties must hold for any cdf: it is a non-decreasing right-continuous function; it tends to 0 at minus infinity and tends to 1 at plus infinity.
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