Introduction¶

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

$Q(k) = \sum_{i > k} p(i)$

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 $$P(1) = p(1), Q(1)=1-p(1)$$.