This minimum value is reached when all the probability density is entirely concentrated at each end " x " = 0 and " x " = 1, with nothing in between : a 2-point Bernoulli distribution with equal probability 1 / 2 at each end ( a coin toss : see section below " Kurtosis bounded by the square of the skewness " for further discussion ).
32.
Since for ? H " ? H " 0 the probability density is concentrated at the two ends " x " = 0 and " x " = 1, this " impossible boundary " is determined by a 2-point distribution : the probability can only take 2 values ( Bernoulli distribution ), one value with probability p and the other with probability " q " = 1 " " p ".
33.
Several authors, including S . Kotz, use the symbols " p " and " q " ( instead of " ? " and " ? " ) for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the Bernoulli distribution, because the beta distribution approaches the Bernoulli distribution in the limit when both shape parameters " ? " and " ? " approach the value of zero.
34.
Several authors, including S . Kotz, use the symbols " p " and " q " ( instead of " ? " and " ? " ) for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the Bernoulli distribution, because the beta distribution approaches the Bernoulli distribution in the limit when both shape parameters " ? " and " ? " approach the value of zero.
35.
At the limit, this is the 2 point Bernoulli distribution with equal probability 1 / 2 at each Dirac delta function end " x " = 0 and " x " = 1 and zero probability everywhere else . ( A coin toss : one face of the coin being " x " = 0 and the other face being " x " = 1 . ) Variance is maximum because the distribution is bimodal with nothing in between the two modes ( spikes ) at each end.
36.
In the absence of such knowledge, some reasonable default values can be used : " For the analyst who prefers simplicity at the cost of some reasonable assumptions, useful prior information can be reduced to an expected model size, an expected " R " 2, and a sample size " ? " determining the weight given to the guess at " R " 2 . " Some researchers suggest the following default values : " R " 2 = 0.5, " ? " = 0.01, and = 0.5 ( parameter of a prior Bernoulli distribution ).
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