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calc: Probability Distribution Functions

 
 9.7 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.  SeeNumerical
 Solutions.
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