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In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).
The probability measure thus defined is known as the Binomial distribution. As we can see from the above formula that, if n=1, the Binomial distribution will turn into a Bernoulli distribution . So we can know that the Bernoulli distribution is exactly a special case of Binomial distribution when n equals to 1.
where Y is a normally distributed random variable with the same expected value and the same variance as X, i.e., E(Y) = np and var(Y) = np(1 − p). This addition of 1/2 to x is a continuity correction.
Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a random vector X. It is defined component by component, as E[X] i = E[X i]. Similarly, one may define the expected value of a random matrix X with components X ij by E[X] ij = E[X ij].
If n and m are large compared to N, and p = m/N is not close to 0 or 1, then X approximately has a Binomial(n, p) distribution. X is a beta-binomial random variable with parameters (n, α, β). Let p = α/(α + β) and suppose α + β is large, then X approximately has a binomial(n, p) distribution.
A Poisson binomial distribution can be approximated by a binomial distribution where , the mean of the , is the success probability of . The variances of P B {\displaystyle PB} and B {\displaystyle B} are related by the formula
There are two methods to define the two-tailed p-value. One method is to sum the probability that the total deviation in numbers of events in either direction from the expected value is either more than or less than the expected value. The probability of that occurring in our example is 0.0437. The second method involves computing the ...
An estimate of the uncertainty in the first and second case can be obtained with the binomial probability distribution using for example the probability of exceedance Pe (i.e. the chance that the event X is larger than a reference value Xr of X) and the probability of non-exceedance Pn (i.e. the chance that the event X is smaller than or equal ...