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Probability Mass Function | The 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 where The following is the plot of the binomial probability density function for four values of p and n = 100. | ||||||||||||||
Cumulative Distribution Function | The formula for the binomial cumulative probability function is The following is the plot of the binomial cumulative distribution function with the same values of p as the pdf plots above. | ||||||||||||||
Percent Point Function | The 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. | ||||||||||||||
Common Statistics |
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Comments | The binomial distribution is probably the most commonly used discrete distribution. | ||||||||||||||
Parameter Estimation | The maximum likelihood estimator of p (n is fixed) is | ||||||||||||||
Software | Most general purpose statistical software programs support at least some of the probability functions for the binomial distribution. |
Learning Objectives Create and interpret frequency polygons Create and interpret cumulative frequency polygons Create and interpret overlaid frequency polygons Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms, but are especially helpful for comparing sets of data. Frequency polygons are also a good choice for displaying cumulative frequency distributions . To create a frequency polygon, start just as for histograms , by choosing a class interval. Then draw an X-axis representing the values of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle value represented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the
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