Why the inverse cumulative probability may not exist or may not be unique

For all continuous distributions handled by inverse cumulative probability, the inverse of the cumulative distribution function (inverse CDF) exists and is unique if 0 < p < 1.

Here are cases when the ICDF may not exist:
  • For all real numbers (for example, normal), inverse CDF is not defined for either p = 0 or p = 1.
  • For all numbers greater than some value (for example, gamma), inverse CDF is defined for p = 0 but not for p = 1.
  • Only on an interval (for example, beta), inverse CDF is defined for p = 0 and p = 1.
Whenever inverse CDF is not defined, Minitab returns a missing value (*) for the result.
For discrete distributions, the situation is more complicated. Suppose you compute the CDF for a binomial with n = 5 and p = 0.4. In this case, there is no value x such that the CDF is 0.5. For x = 1, the CDF is 0.3370; for x = 2, the CDF jumps up to 0.6826.

If the inverse cumulative probabilities are displayed in a table and not stored in a column, both values of x are displayed. If the inverse cumulative probabilities are stored, the larger of the two values is stored in the worksheet column.

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