Next: The Gaussian Distribution, Up: Random Number Distributions [Index]

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 <= P(x) <= 1*, *0 <= Q(x) <= 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 <= 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)*.

Next: The Gaussian Distribution, Up: Random Number Distributions [Index]