# 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.

Whenever inverse CDF is not defined, Minitab returns a missing value (*) for the result, such as in these cases:
• When the probability distribution function (PDF) is positive for the entire real number line (for example, the normal PDF), the ICDF is not defined for either p = 0 or p = 1.
• When the PDF is positive for all values that are greater than some value (for example, the chi-square PDF), the ICDF is defined for p = 0 but not for p = 1.
• When the PDF is positive only on an interval (for example, the uniform PDF), the ICDF is defined for p = 0 and p = 1.
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.
###### Note

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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