## 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 @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.