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 @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.