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

geometric distribution

Physics

The distribution of the number of independent Bernoulli trials before a successful result is obtained; for example, the distribution of the number of times a coin has to be tossed before a head comes up. The probability that the number of trials (x) is k is

$P (x = k) = q^{k - 1} p$

The mean and variance are 1/p and q/p² respectively.

Mathematics

The discrete probability distribution for the number of experiments required to achieve the first success in a sequence of independent experiments, all with the same probability p of success. The probability mass function is given by Pr(X = r) = p(1−p)^r−1, for r = 1, 2,…. It has mean 1/p and variance (1−p)/p².

Statistics

A particular discrete distribution. An experiment or trial with two possible results, usually classified as ‘success’ or ‘failure’, is repeated independently until the first success is obtained. If the probability of a success, p (≠ 0, 1), is the same for each trial, then the distribution of the total number of trials, up to and including the one in which the first success is obtained, is a geometric distribution with parameter p. The probability function is given by $geometric distribution$
The reason for the term ‘geometric’ is that the sequence of probabilities P(X = r), for r = 0, 1, 2,…, forms a geometric progression with first term p and common ratio (1−p). The mean is 1/p and the variance is (1−p)/p². The mode is 1 for all values of p.
The fact that the mode is always 1 means that, even for a very rare event (e.g. a lottery win), the most probable number of trials necessary to obtain the first success is 1. This is sometimes regarded as a paradox.
If we write q = 1−p, cumulative probabilities are given by P(X≥r) = q^r−1 and P(X≤r) = 1−q^r. The geometric distribution is a discrete analogue of the exponential distribution and shares the ‘non-ageing’ or forgetfulness property:
Geometric distribution. All geometric distributions have their mode at the lowest possible value, with successive probabilities having the common ratio p.
$geometric distribution$
If X₁, X₂,…, X_n are independent random variables each having a geometric distribution with parameter p, then , which is the number of trials up to and including the nth success, has a negative binomial distribution with parameters n and p.
Some authors take the definition of a geometric distribution to be the number of trials before the first successful trial. This leads to P(X=r)=p(1−p)^r, for r=0, 1,…, for which the mean is (1−p)/p and the mode is 0, though the variance is unchanged.

Geology and Earth Sciences

See log-normal.

Economics

A discrete distribution with a probability function of the form

$P [X = x | p] = p {(1 - p)}^{x - 1}$

for x = 1,2,…and 0 ≤ p ≤ 1. The geometric distribution has mean 1/p and variance (1− p)/p².

单词	geometric distribution
释义	geometric distribution Physics The distribution of the number of independent Bernoulli trials before a successful result is obtained; for example, the distribution of the number of times a coin has to be tossed before a head comes up. The probability that the number of trials (x) is k is $P (x = k) = q^{k - 1} p$ The mean and variance are 1/p and q/p² respectively. Mathematics The discrete probability distribution for the number of experiments required to achieve the first success in a sequence of independent experiments, all with the same probability p of success. The probability mass function is given by Pr(X = r) = p(1−p)^r−1, for r = 1, 2,…. It has mean 1/p and variance (1−p)/p². Statistics A particular discrete distribution. An experiment or trial with two possible results, usually classified as ‘success’ or ‘failure’, is repeated independently until the first success is obtained. If the probability of a success, p (≠ 0, 1), is the same for each trial, then the distribution of the total number of trials, up to and including the one in which the first success is obtained, is a geometric distribution with parameter p. The probability function is given by $geometric distribution$ The reason for the term ‘geometric’ is that the sequence of probabilities P(X = r), for r = 0, 1, 2,…, forms a geometric progression with first term p and common ratio (1−p). The mean is 1/p and the variance is (1−p)/p². The mode is 1 for all values of p. The fact that the mode is always 1 means that, even for a very rare event (e.g. a lottery win), the most probable number of trials necessary to obtain the first success is 1. This is sometimes regarded as a paradox. If we write q = 1−p, cumulative probabilities are given by P(X≥r) = q^r−1 and P(X≤r) = 1−q^r. The geometric distribution is a discrete analogue of the exponential distribution and shares the ‘non-ageing’ or forgetfulness property: Geometric distribution. All geometric distributions have their mode at the lowest possible value, with successive probabilities having the common ratio p. $geometric distribution$ If X₁, X₂,…, X_n are independent random variables each having a geometric distribution with parameter p, then , which is the number of trials up to and including the nth success, has a negative binomial distribution with parameters n and p. Some authors take the definition of a geometric distribution to be the number of trials before the first successful trial. This leads to P(X=r)=p(1−p)^r, for r=0, 1,…, for which the mean is (1−p)/p and the mode is 0, though the variance is unchanged. Geology and Earth Sciences See log-normal. Economics A discrete distribution with a probability function of the form $P [X = x \| p] = p {(1 - p)}^{x - 1}$ for x = 1,2,…and 0 ≤ p ≤ 1. The geometric distribution has mean 1/p and variance (1− p)/p².
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