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 @c{$\nu_1$}
v1 and @c{$\nu_2$}
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 @c{$\sigma$}
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 @c{$\nu$}
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 @c{$A(t|\nu)$}
A(t|v)
where @c{$A(0|\nu) = 1$}
A(0|v) = 1 and @c{$A(\infty|\nu) \to 0$}
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 section Numerical Solutions.

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