Skip to main content

Binomial distribution


Binomial Distribution

Probability Mass FunctionThe binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled "success" and "failure". The binomial distribution is used to obtain the probability of observing x successes in N trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials.The formula for the binomial probability mass function is
P(x,p,n) = (n x)*(p**x)*(1-p)**(n-x) for x = 0, 1, 2, ..., n
where
(n x) = n!/(x!(n-x)!
The following is the plot of the binomial probability density function for four values of p and n = 100.
plot of the binomial probability density function
Cumulative Distribution FunctionThe formula for the binomial cumulative probability function isF(x,p,n) = SUM[(n i)*(p**i)*(1-p)**(n-i) where the summation is over i from 0 to x
The following is the plot of the binomial cumulative distribution function with the same values of p as the pdf plots above.
plot of the binomial cumulative distribution function
Percent Point FunctionThe binomial percent point function does not exist in simple closed form. It is computed numerically. Note that because this is a discrete distribution that is only defined for integer values of x, the percent point function is not smooth in the way the percent point function typically is for a continuous distribution.The following is the plot of the binomial percent point function with the same values of p as the pdf plots above.
plot of the binomial percent point function
Common Statistics
Meann*p
Modep*(n+1) - 1 <= x <= p*(n+1)
Range0 to N
Standard DeviationSQRT(n*p*(1-p))
Coefficient of VariationSQRT((1-p)/(n*p))
Skewness(1-2*p)/SQRT(n*p*(1-p))
Kurtosis3 - 6/n + 1/(n*p*(1-p))
CommentsThe binomial distribution is probably the most commonly used discrete distribution.
Parameter EstimationThe maximum likelihood estimator of p (n is fixed) isptilde = x/n
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the binomial distribution.

Comments

Post a Comment

Popular posts from this blog

Lognormal distribution

Lognormal Distribution Probability Density Function A variable X is lognormally distributed if Y = LN(X) is normally distributed with "LN" denoting the natural logarithm. The general formula for the  probability density function  of the lognormal distribution is where   is the  shape parameter ,   is the  location parameter  and  m is the  scale parameter . The case where   = 0 and  m  = 1 is called the  standard lognormal distribution . The case where   equals zero is called the 2-parameter lognormal distribution. The equation for the standard lognormal distribution is Since the general form of probability functions can be  expressed in terms of the standard distribution , all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the lognormal probability density function for four values of  . There are several commo...
Post a Comment
Read more

Runs Test for Detecting Non-randomness

Runs Test for Detecting Non-randomness Purpose: Detect Non-Randomness The runs test ( Bradley, 1968 ) can be used to decide if a data set is from a random process. A run is defined as a series of increasing values or a series of decreasing values. The number of increasing, or decreasing, values is the length of the run. In a random data set, the probability that the ( I +1)th value is larger or smaller than the I th value follows a binomial distribution , which forms the basis of the runs test. Typical Analysis and Test Statistics The first step in the runs test is to count the number of runs in the data sequence. There are several ways to define runs in the literature, however, in all cases the formulation must produce a dichotomous sequence of values. For example, a series of 20 coin tosses might produce the f...
Post a Comment
Read more

Poisson distribution

Poisson Distribution Probability Mass Function The Poisson distribution is used to model the number of events occurring within a given time interval.The formula for the Poisson probability mass function is  is the shape parameter which indicates the average number of events in the given time interval. The following is the plot of the Poisson probability density function for four values of  . Cumulative Distribution Function The formula for the Poisson cumulative probability function is The following is the plot of the Poisson cumulative distribution function with the same values of   as the pdf plots above. Percent Point Function The Poisson percent point function does not exist in simple closed form. It is computed numerically. Note that because this is a discrete distribution that is only defined for integer values of  x , the percent point function is not smooth in the way the percent point function typically is for a continuous distribution....
Post a Comment
Read more