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-utp``x`

) [`utp``x`

]
and `ltp``x`

], for various letters
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.