Probability Distribution Functions
==================================

The functions in this section compute various probability distributions.
For continuous distributions, this is the integral of the probability
density function from `x' to infinity.  (These are the "upper
tail" distribution functions; there are also corresponding "lower
tail" functions which integrate from minus infinity to `x'.)
For discrete distributions, the upper tail function gives the sum
from `x' to infinity; the lower tail function gives the sum
from minus infinity up to, but not including, `x'.

To integrate from `x' to `y', just use the distribution function twice
and subtract.  For example, the probability that a Gaussian random
variable with mean 2 and standard deviation 1 will lie in the range
from 2.5 to 2.8 is `utpn(2.5,2,1) - utpn(2.8,2,1)' ("the probability
that it is greater than 2.5, but not greater than 2.8"), or
equivalently `ltpn(2.8,2,1) - ltpn(2.5,2,1)'.

The `k B' (`calc-utpb') [`utpb'] function uses the binomial
distribution.  Push the parameters N, P, and then X onto the stack;
the result (`utpb(x,n,p)') is the probability that an event will occur
X or more times out of N trials, if its probability of occurring in
any given trial is P.  The `I k B' [`ltpb'] function is the
probability that the event will occur fewer than X times.

The other probability distribution functions similarly take the form
`k X' (`calc-utpX') [`utpX'] and `I k X' [`ltpX'], for various letters
X.  The arguments to the algebraic functions are the value of the
random variable first, then whatever other parameters define the
distribution.  Note these are among the few Calc functions where the
order of the arguments in algebraic form differs from the order of
arguments as found on the stack.  (The random variable comes last on
the stack, so that you can type, e.g., `2 RET 1 RET 2.5 k N M-RET DEL
2.8 k N -', using `M-RET DEL' to recover the original arguments but
substitute a new value for `x'.)

The `utpc(x,v)' function uses the chi-square distribution with `v'
degrees of freedom.  It is the probability that a model is correct if
its chi-square statistic is `x'.

The `utpf(F,v1,v2)' function uses the F distribution, used in various
statistical tests.  The parameters `v1' and `v2' are the degrees of
freedom in the numerator and denominator, respectively, used in
computing the statistic `F'.

The `utpn(x,m,s)' function uses a normal (Gaussian) distribution with
mean `m' and standard deviation `s'.  It is the probability that such
a normal-distributed random variable would exceed `x'.

The `utpp(n,x)' function uses a Poisson distribution with mean `x'.
It is the probability that `n' or more such Poisson random events will
occur.

The `utpt(t,v)' function uses the Student's "t" distribution with `v'
degrees of freedom.  It is the probability that a t-distributed random
variable will be greater than `t'.  (Note: This computes the
distribution function `A(t|v)' where `A(0|v) = 1' and `A(inf|v) -> 0'.
The `UTPT' operation on the HP-48 uses a different definition which
returns half of Calc's value: `UTPT(t,v) = .5*utpt(t,v)'.)

While Calc does not provide inverses of the probability distribution
functions, the `a R' command can be used to solve for the inverse.
Since the distribution functions are monotonic, `a R' is guaranteed
to be able to find a solution given any initial guess.
See Numerical Solutions.