IntroductionΒΆ

Continuous random number distributions are defined by a probability density function, \(p(x)\), such that the probability of \(x\) occurring in the infinitesimal range \(x\) to \(x+dx\) is \(p dx\).

The cumulative distribution function for the lower tail \(P(x)\) is defined by the integral,

\[P(x) = \int_{-\infty}^{x} dx' p(x')\]

and gives the probability of a variate taking a value less than \(x\).

The cumulative distribution function for the upper tail \(Q(x)\) is defined by the integral,

\[Q(x) = \int_{x}^{+\infty} dx' p(x')\]

and gives the probability of a variate taking a value greater than \(x\).

The upper and lower cumulative distribution functions are related by \(P(x) + Q(x) = 1\) and satisfy \(0 \leq P(x) \leq 1, 0 \leq Q(x) \leq 1\).

The inverse cumulative distributions, \(x=P^{-1}(P)\) and \(x=Q^{-1}(Q)\) give the values of \(x\) which correspond to a specific value of \(P\) or \(Q\). They can be used to find confidence limits from probability values.

For discrete distributions the probability of sampling the integer value \(k\) is given by \(p(k)\), where \(\sum_k p(k) = 1\). The cumulative distribution for the lower tail \(P(k)\) of a discrete distribution is defined as,

\[P(k) = \sum_{i \leq k} p(i)\]

where the sum is over the allowed range of the distribution less than or equal to \(k\).

The cumulative distribution for the upper tail of a discrete distribution \(Q(k)\) is defined as

\[\begin{split}Q(k) = \sum_{i > k} p(i)\end{split}\]

giving the sum of probabilities for all values greater than \(k\). These two definitions satisfy the identity \(P(k)+Q(k)=1\).

If the range of the distribution is \(1\) to \(n\) inclusive then \(P(n)=1, Q(n)=0\) while \(P(1) = p(1), Q(1)=1-p(1)\).