The mode is (a) any value in (0,1) for α, β = 1, (b) 0 or 1 for α, β 1 and (d) 1 if β ≤ 1 and α > 1. Key statistical properties of the beta distribution are:įigure 3 – Statistical properties of the beta distribution ChartĪ chart of the beta distribution for β = 8 and α = 2, 4 and 6 is displayed in Figure 1.Īctually, the beta distribution can take a great variety of shapes, as shown in Figure 2.įigure 2 – Other shapes of beta distribution Properties Note that in the general case, α + β does not have to be a positive integer, although α and β do have to be positive numbers and x must be between 0 and 1. This is a special case of the pdf of the beta distribution The probability density function (pdf) for x = the probability of success on any single trial is given by Let α = # of successes in n trials and β = # of failures in n trials (and so α + β = n). Instead, we would now like to view the probability of success on any single trial as the random variable, and the number of trials n and the total number of successes in n trials as constants. Definition 1: For the binomial distribution the number of successes x is a random variable and the number of trials n and the probability of success p on any single trial are parameters (i.e.
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