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

binomial distribution

Physics

The distribution of the number of successes in an experiment in which there are two possible outcomes, success and failure. The probability of k successes is:

$b (k, n, p) = n! / k! (n - k)! \times p^{n} \times q^{n - k}$

where p is the probability of success and q (= 1−p) the probability of failure on each trial. These probabilities are given by the terms in the binomial expansion of (p+q)ⁿ (see binomial theorem). The distribution has a mean np and variance npq. If n is large and p small it can be approximated by a Poisson distribution with mean np. If n is large and p is not near 0 or 1, it can be approximated by a normal distribution with mean np and variance npq. See also Pascal’s distribution.

Mathematics

The discrete probability distribution for the number of successes when n independent Bernoulli trials are carried out, each with the same probability p of success. The probability mass function is given by $Pr (X = r) =^{n} C_{r} p^{r} {(1 - p)}^{n - r}$ , for 0 ≤ r ≤ n. This distribution is denoted B(n, p) and has mean np and variance np(1−p).

Statistics

The distribution associated with the random variable, X, defined as the number of ‘successes’ in n independent trials each having the same probability, p, of success. The random variable X is said to be a binomial variable and to have a binomial distribution with parameters n and p. This is written as X ~ B(n, p). The mean of this distribution is np and the variance is np (1−p). The probability function is given by $binomial distribution$ The distribution takes its name from the fact that successive probabilities are the terms in the expansion in ascending powers of p, by the binomial theorem, of (q+p)ⁿ, where q=1−p. The first published derivation of the distribution was by Jacob Bernoulli in 1713.
As an example, suppose that a computer generates fifteen random integers between 0 and 9 inclusive. The number of these integers that are odd has a B(15, 0.5) distribution. The number that are non-zero has a B(15, 0.9) distribution, and the number that are greater than 7 has a B(15, 0.2) distribution. The diagram shows the graphs of the probability functions for these distributions.
If we note that P(X=0)=qⁿ, successive probabilities can be calculated using the recurrence relation $binomial distribution$ If (n+1)p is not an integer the graph is unimodal, with mode at the (integer) value of r such that and (n+1)p−1 and (n+1)p are both modal values (as in the B(15, 0.5) case illustrated).
A binomial random variable with parameters n and p may be regarded as the sum of n independent observations of a Bernoulli variable (see Bernoulli distribution) with parameter p. The sum of two independent binomial variables with parameters n₁, p and n₂, p, respectively, is also a binomial variable, with parameters (n₁+n₂), p.
For large values of np and nq the normal approximation to the binomial distribution may be used: $binomial distribution$ and Φ is the cumulative distribution function for a standard normal variable (see normal distribution). The ‘½’ is a continuity correction. The result, that a binomial distribution with p = ½ may be approximated by a normal distribution, underlies the derivation of the normal distribution by de Moivre in 1733 and is sometimes referred to as the de Moivre–Laplace theorem. For large values of n and small values of p the Poisson approximation to the binomial distribution may be used: $binomial distribution$ The word ‘binomial’ was used in its mathematical sense in a 1557 text entitled The Whetstone of Witte by Robert Recorde. The ‘binomial distribution’ was so named by Yule in 1911.
http://www.math.uah.edu/stat/applets/BinomialCoinExperiment.html Applet.
Binomial distribution. The distribution is skewed to the right if p>0.5 and to the left if p<0.5. It is symmetric if p=0.5.

Computer

The basic discrete probability distribution for data in the form of proportions. An event, E, can occur with probability p. In a sample of n independent trials the probability that E occurs exactly r times is
$^{n} C_{r} p^{r} {(1 - p)}^{n - r}$
(see combination). The distribution is discrete, taking only the values 0, 1, 2,…, n. The mean of the binomial distribution is np and the variance is
$n p / (1 - p)$

Geology and Earth Sciences

In statistics, a discrete probability distribution of the number of outcomes of a particular kind occurring for a set number of trials, where one of two outcomes is possible, each trial is independent, and the probability of a particular outcome is constant.

Economics

The distribution of the number of occurrences of a random event as the result of making a number of independent draws, with a known and constant probability of the event occurring each time. If the probability of the event (for example ‘heads’ throwing a coin, or ‘6’ throwing a die) each time is p, and the probability of non-occurrence is (1 − p), the binomial distribution gives the probability of exactly r occurrences in n independent tries, where the r satisfies 0 ≤ r ≤ n. This probability is given by

$p^{r} {(1 - p)}^{n - r}_{n} C_{r}$

_nC_r denotes the number of ways of choosing r objects out of n and is called the binomial coefficient

$_{n} C_{r} = n! / [r! (n - r)!]$

where r! denotes ‘r factorial’, defined as 0! = 1 and for r > 0

$r! = r (r - 1) (r - 2) … (2) (1) .$

单词	binomial distribution
释义	binomial distribution Physics The distribution of the number of successes in an experiment in which there are two possible outcomes, success and failure. The probability of k successes is: $b (k, n, p) = n! / k! (n - k)! \times p^{n} \times q^{n - k}$ where p is the probability of success and q (= 1−p) the probability of failure on each trial. These probabilities are given by the terms in the binomial expansion of (p+q)ⁿ (see binomial theorem). The distribution has a mean np and variance npq. If n is large and p small it can be approximated by a Poisson distribution with mean np. If n is large and p is not near 0 or 1, it can be approximated by a normal distribution with mean np and variance npq. See also Pascal’s distribution. Mathematics The discrete probability distribution for the number of successes when n independent Bernoulli trials are carried out, each with the same probability p of success. The probability mass function is given by $Pr (X = r) =^{n} C_{r} p^{r} {(1 - p)}^{n - r}$ , for 0 ≤ r ≤ n. This distribution is denoted B(n, p) and has mean np and variance np(1−p). Statistics The distribution associated with the random variable, X, defined as the number of ‘successes’ in n independent trials each having the same probability, p, of success. The random variable X is said to be a binomial variable and to have a binomial distribution with parameters n and p. This is written as X ~ B(n, p). The mean of this distribution is np and the variance is np (1−p). The probability function is given by $binomial distribution$ The distribution takes its name from the fact that successive probabilities are the terms in the expansion in ascending powers of p, by the binomial theorem, of (q+p)ⁿ, where q=1−p. The first published derivation of the distribution was by Jacob Bernoulli in 1713. As an example, suppose that a computer generates fifteen random integers between 0 and 9 inclusive. The number of these integers that are odd has a B(15, 0.5) distribution. The number that are non-zero has a B(15, 0.9) distribution, and the number that are greater than 7 has a B(15, 0.2) distribution. The diagram shows the graphs of the probability functions for these distributions. If we note that P(X=0)=qⁿ, successive probabilities can be calculated using the recurrence relation $binomial distribution$ If (n+1)p is not an integer the graph is unimodal, with mode at the (integer) value of r such that and (n+1)p−1 and (n+1)p are both modal values (as in the B(15, 0.5) case illustrated). A binomial random variable with parameters n and p may be regarded as the sum of n independent observations of a Bernoulli variable (see Bernoulli distribution) with parameter p. The sum of two independent binomial variables with parameters n₁, p and n₂, p, respectively, is also a binomial variable, with parameters (n₁+n₂), p. For large values of np and nq the normal approximation to the binomial distribution may be used: $binomial distribution$ and Φ is the cumulative distribution function for a standard normal variable (see normal distribution). The ‘½’ is a continuity correction. The result, that a binomial distribution with p = ½ may be approximated by a normal distribution, underlies the derivation of the normal distribution by de Moivre in 1733 and is sometimes referred to as the de Moivre–Laplace theorem. For large values of n and small values of p the Poisson approximation to the binomial distribution may be used: $binomial distribution$ The word ‘binomial’ was used in its mathematical sense in a 1557 text entitled The Whetstone of Witte by Robert Recorde. The ‘binomial distribution’ was so named by Yule in 1911. http://www.math.uah.edu/stat/applets/BinomialCoinExperiment.html Applet. Binomial distribution. The distribution is skewed to the right if p>0.5 and to the left if p<0.5. It is symmetric if p=0.5. Computer The basic discrete probability distribution for data in the form of proportions. An event, E, can occur with probability p. In a sample of n independent trials the probability that E occurs exactly r times is $^{n} C_{r} p^{r} {(1 - p)}^{n - r}$ (see combination). The distribution is discrete, taking only the values 0, 1, 2,…, n. The mean of the binomial distribution is np and the variance is $n p / (1 - p)$ Geology and Earth Sciences In statistics, a discrete probability distribution of the number of outcomes of a particular kind occurring for a set number of trials, where one of two outcomes is possible, each trial is independent, and the probability of a particular outcome is constant. Economics The distribution of the number of occurrences of a random event as the result of making a number of independent draws, with a known and constant probability of the event occurring each time. If the probability of the event (for example ‘heads’ throwing a coin, or ‘6’ throwing a die) each time is p, and the probability of non-occurrence is (1 − p), the binomial distribution gives the probability of exactly r occurrences in n independent tries, where the r satisfies 0 ≤ r ≤ n. This probability is given by $p^{r} {(1 - p)}^{n - r}_{n} C_{r}$ _nC_r denotes the number of ways of choosing r objects out of n and is called the binomial coefficient $_{n} C_{r} = n! / [r! (n - r)!]$ where r! denotes ‘r factorial’, defined as 0! = 1 and for r > 0 $r! = r (r - 1) (r - 2) … (2) (1) .$
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