

10. Scientific Functions

The functions described here perform trigonometric and other transcendental calculations. They generally produce floating-point answers correct to the full current precision. The H (Hyperbolic) and I (Inverse) flag keys must be used to get some of these functions from the keyboard.

One miscellanous command is shift-P (calc-pi), which pushes the value of pi (at the current precision) onto the stack. With the Hyperbolic flag, it pushes the value e, the base of natural logarithms. With the Inverse flag, it pushes Euler's constant gamma (about 0.5772). With both Inverse and Hyperbolic, it pushes the "golden ratio" phi (about 1.618). (At present, Euler's constant is not available to unlimited precision; Calc knows only the first 100 digits.) In Symbolic mode, these commands push the actual variables `pi', `e', `gamma', and `phi', respectively, instead of their values; see section 8.4.5 Symbolic Mode.

The Q (calc-sqrt) [sqrt] function is described elsewhere; see section 9.1 Basic Arithmetic. With the Inverse flag [sqr], this command computes the square of the argument.

See section 5.7 Numeric Prefix Arguments, for a discussion of the effect of numeric prefix arguments on commands in this chapter which do not otherwise interpret a prefix argument.

10.1 Logarithmic Functions  
10.2 Trigonometric/Hyperbolic Functions  
10.3 Advanced Mathematical Functions  
10.4 Branch Cuts and Principal Values  
10.5 Random Numbers  
10.6 Combinatorial Functions  
10.7 Probability Distribution Functions  


10.1 Logarithmic Functions

The shift-L (calc-ln) [ln] command computes the natural logarithm of the real or complex number on the top of the stack. With the Inverse flag it computes the exponential function instead, although this is redundant with the E command.

The shift-E (calc-exp) [exp] command computes the exponential, i.e., e raised to the power of the number on the stack. The meanings of the Inverse and Hyperbolic flags follow from those for the calc-ln command.

The H L (calc-log10) [log10] command computes the common (base-10) logarithm of a number. (With the Inverse flag [exp10], it raises ten to a given power.) Note that the common logarithm of a complex number is computed by taking the natural logarithm and dividing by ln(10).

The B (calc-log) [log] command computes a logarithm to any base. For example, 1024 RET 2 B produces 10, since 2^10 = 1024. In certain cases like `log(3,9)', the result will be either 1:2 or 0.5 depending on the current Fraction Mode setting. With the Inverse flag [alog], this command is similar to ^ except that the order of the arguments is reversed.

The f I (calc-ilog) [ilog] command computes the integer logarithm of a number to any base. The number and the base must themselves be positive integers. This is the true logarithm, rounded down to an integer. Thus ilog(x,10) is 3 for all x in the range from 1000 to 9999. If both arguments are positive integers, exact integer arithmetic is used; otherwise, this is equivalent to `floor(log(x,b))'.

The f E (calc-expm1) [expm1] command computes exp(x)-1, but using an algorithm that produces a more accurate answer when the result is close to zero, i.e., when exp(x) is close to one.

The f L (calc-lnp1) [lnp1] command computes ln(x+1), producing a more accurate answer when x is close to zero.


10.2 Trigonometric/Hyperbolic Functions


Calc's "units" mechanism includes angular units like deg, rad, and grad. While `sin(45 deg)' is not evaluated all the time, the u s (calc-simplify-units) command will simplify `sin(45 deg)' by taking the sine of 45 degrees, regardless of the current angular mode. See section 13.1 Basic Operations on Units.

Also, the symbolic variable pi is not ordinarily recognized in arguments to trigonometric functions, as in `sin(3 pi / 4)', but the a s (calc-simplify) command recognizes many such formulas when the current angular mode is radians and symbolic mode is enabled; this example would be replaced by `sqrt(2) / 2'. See section 8.4.5 Symbolic Mode. Beware, this simplification occurs even if you have stored a different value in the variable `pi'; this is one reason why changing built-in variables is a bad idea. Arguments of the form x plus a multiple of pi/2 are also simplified. Calc includes similar formulas for cos and tan.

The a s command knows all angles which are integer multiples of pi/12, pi/10, or pi/8 radians. In degrees mode, analogous simplifications occur for integer multiples of 15 or 18 degrees, and for arguments plus multiples of 90 degrees.





The calc-sincos [sincos] command computes the sine and cosine of a number, returning them as a vector of the form `[cos, sin]'. With the Inverse flag [arcsincos], this command takes a two-element vector as an argument and computes arctan2 of the elements. (This command does not accept the Hyperbolic flag.)


10.3 Advanced Mathematical Functions

Calc can compute a variety of less common functions that arise in various branches of mathematics. All of the functions described in this section allow arbitrary complex arguments and, except as noted, will work to arbitrarily large precisions. They can not at present handle error forms or intervals as arguments.

NOTE: These functions are still experimental. In particular, their accuracy is not guaranteed in all domains. It is advisable to set the current precision comfortably higher than you actually need when using these functions. Also, these functions may be impractically slow for some values of the arguments.

The f g (calc-gamma) [gamma] command computes the Euler gamma function. For positive integer arguments, this is related to the factorial function: `gamma(n+1) = fact(n)'. For general complex arguments the gamma function can be defined by the following definite integral: gamma(a) = integ(t^(a-1) exp(t), t, 0, inf). (The actual implementation uses far more efficient computational methods.)

The f G (calc-inc-gamma) [gammaP] command computes the incomplete gamma function, denoted `P(a,x)'. This is defined by the integral, gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a). This implies that `gammaP(a,inf) = 1' for any a (see the definition of the normal gamma function).

Several other varieties of incomplete gamma function are defined. The complement of P(a,x), called Q(a,x) = 1-P(a,x) by some authors, is computed by the I f G [gammaQ] command. You can think of this as taking the other half of the integral, from x to infinity.

The functions corresponding to the integrals that define P(a,x) and Q(a,x) but without the normalizing 1/gamma(a) factor are called g(a,x) and G(a,x), respectively (where g and G represent the lower- and upper-case Greek letter gamma). You can obtain these using the H f G [gammag] and H I f G [gammaG] commands.

The f b (calc-beta) [beta] command computes the Euler beta function, which is defined in terms of the gamma function as beta(a,b) = gamma(a) gamma(b) / gamma(a+b), or by beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1).

The f B (calc-inc-beta) [betaI] command computes the incomplete beta function I(x,a,b). It is defined by betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b). Once again, the H (hyperbolic) prefix gives the corresponding un-normalized version [betaB].

The f e (calc-erf) [erf] command computes the error function erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi). The complementary error function I f e (calc-erfc) [erfc] is the corresponding integral from `x' to infinity; the sum erf(x) + erfc(x) = 1.

The f j (calc-bessel-J) [besJ] and f y (calc-bessel-Y) [besY] commands compute the Bessel functions of the first and second kinds, respectively. In `besJ(n,x)' and `besY(n,x)' the "order" parameter n is often an integer, but is not required to be one. Calc's implementation of the Bessel functions currently limits the precision to 8 digits, and may not be exact even to that precision. Use with care!


10.4 Branch Cuts and Principal Values

All of the logarithmic, trigonometric, and other scientific functions are defined for complex numbers as well as for reals. This section describes the values returned in cases where the general result is a family of possible values. Calc follows section 12.5.3 of Steele's Common Lisp, the Language, second edition, in these matters. This section will describe each function briefly; for a more detailed discussion (including some nifty diagrams), consult Steele's book.

Note that the branch cuts for arctan and arctanh were changed between the first and second editions of Steele. Versions of Calc starting with 2.00 follow the second edition.

The new branch cuts exactly match those of the HP-28/48 calculators. They also match those of Mathematica 1.2, except that Mathematica's arctan cut is always in the right half of the complex plane, and its arctanh cut is always in the top half of the plane. Calc's cuts are continuous with quadrants I and III for arctan, or II and IV for arctanh.

Note: The current implementations of these functions with complex arguments are designed with proper behavior around the branch cuts in mind, not efficiency or accuracy. You may need to increase the floating precision and wait a while to get suitable answers from them.

For `sqrt(a+bi)': When a<0 and b is small but positive or zero, the result is close to the +i axis. For b small and negative, the result is close to the -i axis. The result always lies in the right half of the complex plane.

For `ln(a+bi)': The real part is defined as `ln(abs(a+bi))'. The imaginary part is defined as `arg(a+bi) = arctan2(b,a)'. Thus the branch cuts for sqrt and ln both lie on the negative real axis.

The following table describes these branch cuts in another way. If the real and imaginary parts of z are as shown, then the real and imaginary parts of f(z) will be as shown. Here eps stands for a small positive value; each occurrence of eps may stand for a different small value.

     z           sqrt(z)       ln(z)
   +,   0         +,  0       any, 0
   -,   0         0,  +       any, pi
   -, +eps      +eps, +      +eps, +
   -, -eps      +eps, -      +eps, -

For `z1^z2': This is defined by `exp(ln(z1)*z2)'. One interesting consequence of this is that `(-8)^1:3' does not evaluate to -2 as you might expect, but to the complex number (1., 1.732). Both of these are valid cube roots of -8 (as is (1., -1.732)); Calc chooses a perhaps less-obvious root for the sake of mathematical consistency.

For `arcsin(z)': This is defined by `-i*ln(i*z + sqrt(1-z^2))'. The branch cuts are on the real axis, less than -1 and greater than 1.

For `arccos(z)': This is defined by `-i*ln(z + i*sqrt(1-z^2))', or equivalently by `pi/2 - arcsin(z)'. The branch cuts are on the real axis, less than -1 and greater than 1.

For `arctan(z)': This is defined by `(ln(1+i*z) - ln(1-i*z)) / (2*i)'. The branch cuts are on the imaginary axis, below -i and above i.

For `arcsinh(z)': This is defined by `ln(z + sqrt(1+z^2))'. The branch cuts are on the imaginary axis, below -i and above i.

For `arccosh(z)': This is defined by `ln(z + (z+1)*sqrt((z-1)/(z+1)))'. The branch cut is on the real axis less than 1.

For `arctanh(z)': This is defined by `(ln(1+z) - ln(1-z)) / 2'. The branch cuts are on the real axis, less than -1 and greater than 1.

The following tables for arcsin, arccos, and arctan assume the current angular mode is radians. The hyperbolic functions operate independently of the angular mode.

       z             arcsin(z)            arccos(z)
 (-1..1),  0      (-pi/2..pi/2), 0       (0..pi), 0
 (-1..1), +eps    (-pi/2..pi/2), +eps    (0..pi), -eps
 (-1..1), -eps    (-pi/2..pi/2), -eps    (0..pi), +eps
   <-1,    0          -pi/2,     +         pi,    -
   <-1,  +eps      -pi/2 + eps,  +      pi - eps, -
   <-1,  -eps      -pi/2 + eps,  -      pi - eps, +
    >1,    0           pi/2,     -          0,    +
    >1,  +eps       pi/2 - eps,  +        +eps,   -
    >1,  -eps       pi/2 - eps,  -        +eps,   +

       z            arccosh(z)         arctanh(z)
 (-1..1),  0        0,  (0..pi)       any,     0
 (-1..1), +eps    +eps, (0..pi)       any,    +eps
 (-1..1), -eps    +eps, (-pi..0)      any,    -eps
   <-1,    0        +,    pi           -,     pi/2
   <-1,  +eps       +,  pi - eps       -,  pi/2 - eps
   <-1,  -eps       +, -pi + eps       -, -pi/2 + eps
    >1,    0        +,     0           +,    -pi/2
    >1,  +eps       +,   +eps          +,  pi/2 - eps
    >1,  -eps       +,   -eps          +, -pi/2 + eps

       z           arcsinh(z)           arctan(z)
   0, (-1..1)    0, (-pi/2..pi/2)         0,     any
   0,   <-1      -,    -pi/2            -pi/2,    -
 +eps,  <-1      +, -pi/2 + eps       pi/2 - eps, -
 -eps,  <-1      -, -pi/2 + eps      -pi/2 + eps, -
   0,    >1      +,     pi/2             pi/2,    +
 +eps,   >1      +,  pi/2 - eps       pi/2 - eps, +
 -eps,   >1      -,  pi/2 - eps      -pi/2 + eps, +

Finally, the following identities help to illustrate the relationship between the complex trigonometric and hyperbolic functions. They are valid everywhere, including on the branch cuts.

sin(i*z)  = i*sinh(z)       arcsin(i*z)  = i*arcsinh(z)
cos(i*z)  =   cosh(z)       arcsinh(i*z) = i*arcsin(z)
tan(i*z)  = i*tanh(z)       arctan(i*z)  = i*arctanh(z)
sinh(i*z) = i*sin(z)        cosh(i*z)    =   cos(z)

The "advanced math" functions (gamma, Bessel, etc.) are also defined for general complex arguments, but their branch cuts and principal values are not rigorously specified at present.


10.5 Random Numbers

The k r (calc-random) [random] command produces random numbers of various sorts.

Given a positive numeric prefix argument M, it produces a random integer N in the range 0 <= N < M. Each of the M values appears with equal probability.

With no numeric prefix argument, the k r command takes its argument from the stack instead. Once again, if this is a positive integer M the result is a random integer less than M. However, note that while numeric prefix arguments are limited to six digits or so, an M taken from the stack can be arbitrarily large. If M is negative, the result is a random integer in the range M < N <= 0.

If the value on the stack is a floating-point number M, the result is a random floating-point number N in the range 0 <= N < M or M < N <= 0, according to the sign of M.

If M is zero, the result is a Gaussian-distributed random real number; the distribution has a mean of zero and a standard deviation of one. The algorithm used generates random numbers in pairs; thus, every other call to this function will be especially fast.

If M is an error form `m +/- s' where m and s are both real numbers, the result uses a Gaussian distribution with mean m and standard deviation s.

If M is an interval form, the lower and upper bounds specify the acceptable limits of the random numbers. If both bounds are integers, the result is a random integer in the specified range. If either bound is floating-point, the result is a random real number in the specified range. If the interval is open at either end, the result will be sure not to equal that end value. (This makes a big difference for integer intervals, but for floating-point intervals it's relatively minor: with a precision of 6, `random([1.0..2.0))' will return any of one million numbers from 1.00000 to 1.99999; `random([1.0..2.0])' may additionally return 2.00000, but the probability of this happening is extremely small.)

If M is a vector, the result is one element taken at random from the vector. All elements of the vector are given equal probabilities.

The sequence of numbers produced by k r is completely random by default, i.e., the sequence is seeded each time you start Calc using the current time and other information. You can get a reproducible sequence by storing a particular "seed value" in the Calc variable RandSeed. Any integer will do for a seed; integers of from 1 to 12 digits are good. If you later store a different integer into RandSeed, Calc will switch to a different pseudo-random sequence. If you "unstore" RandSeed, Calc will re-seed itself from the current time. If you store the same integer that you used before back into RandSeed, you will get the exact same sequence of random numbers as before.

The calc-rrandom command (not on any key) produces a random real number between zero and one. It is equivalent to `random(1.0)'.

The k a (calc-random-again) command produces another random number, re-using the most recent value of M. With a numeric prefix argument n, it produces n more random numbers using that value of M.

The k h (calc-shuffle) command produces a vector of several random values with no duplicates. The value on the top of the stack specifies the set from which the random values are drawn, and may be any of the M formats described above. The numeric prefix argument gives the length of the desired list. (If you do not provide a numeric prefix argument, the length of the list is taken from the top of the stack, and M from second-to-top.)

If M is a floating-point number, zero, or an error form (so that the random values are being drawn from the set of real numbers) there is little practical difference between using k h and using k r several times. But if the set of possible values consists of just a few integers, or the elements of a vector, then there is a very real chance that multiple k r's will produce the same number more than once. The k h command produces a vector whose elements are always distinct. (Actually, there is a slight exception: If M is a vector, no given vector element will be drawn more than once, but if several elements of M are equal, they may each make it into the result vector.)

One use of k h is to rearrange a list at random. This happens if the prefix argument is equal to the number of values in the list: [1, 1.5, 2, 2.5, 3] 5 k h might produce the permuted list `[2.5, 1, 1.5, 3, 2]'. As a convenient feature, if the argument n is negative it is replaced by the size of the set represented by M. Naturally, this is allowed only when M specifies a small discrete set of possibilities.

To do the equivalent of k h but with duplications allowed, given M on the stack and with n just entered as a numeric prefix, use v b to build a vector of copies of M, then use V M k r to "map" the normal k r function over the elements of this vector. See section 11. Vector/Matrix Functions.

10.5.1 Random Number Generator  (Complete description of Calc's algorithm)


10.5.1 Random Number Generator

Calc's random number generator uses several methods to ensure that the numbers it produces are highly random. Knuth's Art of Computer Programming, Volume II, contains a thorough description of the theory of random number generators and their measurement and characterization.

If RandSeed has no stored value, Calc calls Emacs' built-in random function to get a stream of random numbers, which it then treats in various ways to avoid problems inherent in the simple random number generators that many systems use to implement random.

When Calc's random number generator is first invoked, it "seeds" the low-level random sequence using the time of day, so that the random number sequence will be different every time you use Calc.

Since Emacs Lisp doesn't specify the range of values that will be returned by its random function, Calc exercises the function several times to estimate the range. When Calc subsequently uses the random function, it takes only 10 bits of the result near the most-significant end. (It avoids at least the bottom four bits, preferably more, and also tries to avoid the top two bits.) This strategy works well with the linear congruential generators that are typically used to implement random.

If RandSeed contains an integer, Calc uses this integer to seed an "additive congruential" method (Knuth's algorithm 3.2.2A, computing X_n-55 - X_n-24). This method expands the seed value into a large table which is maintained internally; the variable RandSeed is changed from, e.g., 42 to the vector [42] to indicate that the seed has been absorbed into this table. When RandSeed contains a vector, k r and related commands continue to use the same internal table as last time. There is no way to extract the complete state of the random number generator so that you can restart it from any point; you can only restart it from the same initial seed value. A simple way to restart from the same seed is to type s r RandSeed to get the seed vector, v u to unpack it back into a number, then s t RandSeed to reseed the generator with that number.

Calc uses a "shuffling" method as described in algorithm 3.2.2B of Knuth. It fills a table with 13 random 10-bit numbers. Then, to generate a new random number, it uses the previous number to index into the table, picks the value it finds there as the new random number, then replaces that table entry with a new value obtained from a call to the base random number generator (either the additive congruential generator or the random function supplied by the system). If there are any flaws in the base generator, shuffling will tend to even them out. But if the system provides an excellent random function, shuffling will not damage its randomness.

To create a random integer of a certain number of digits, Calc builds the integer three decimal digits at a time. For each group of three digits, Calc calls its 10-bit shuffling random number generator (which returns a value from 0 to 1023); if the random value is 1000 or more, Calc throws it out and tries again until it gets a suitable value.

To create a random floating-point number with precision p, Calc simply creates a random p-digit integer and multiplies by 10^-p. The resulting random numbers should be very clean, but note that relatively small numbers will have few significant random digits. In other words, with a precision of 12, you will occasionally get numbers on the order of 10^-9 or 10^-10, but those numbers will only have two or three random digits since they correspond to small integers times 10^-12.

To create a random integer in the interval `[0 .. m)', Calc counts the digits in m, creates a random integer with three additional digits, then reduces modulo m. Unless m is a power of ten the resulting values will be very slightly biased toward the lower numbers, but this bias will be less than 0.1%. (For example, if m is 42, Calc will reduce a random integer less than 100000 modulo 42 to get a result less than 42. It is easy to show that the numbers 40 and 41 will be only 2380/2381 as likely to result from this modulo operation as numbers 39 and below.) If m is a power of ten, however, the numbers should be completely unbiased.

The Gaussian random numbers generated by `random(0.0)' use the "polar" method described in Knuth section 3.4.1C. This method generates a pair of Gaussian random numbers at a time, so only every other call to `random(0.0)' will require significant calculations.


10.6 Combinatorial Functions

Commands relating to combinatorics and number theory begin with the k key prefix.

The k g (calc-gcd) [gcd] command computes the Greatest Common Divisor of two integers. It also accepts fractions; the GCD of two fractions is defined by taking the GCD of the numerators, and the LCM of the denominators. This definition is consistent with the idea that `a / gcd(a,x)' should yield an integer for any `a' and `x'. For other types of arguments, the operation is left in symbolic form.

The k l (calc-lcm) [lcm] command computes the Least Common Multiple of two integers or fractions. The product of the LCM and GCD of two numbers is equal to the product of the numbers.

The k E (calc-extended-gcd) [egcd] command computes the GCD of two integers x and y and returns a vector [g, a, b] where g = gcd(x,y) = a x + b y.

The ! (calc-factorial) [fact] command computes the factorial of the number at the top of the stack. If the number is an integer, the result is an exact integer. If the number is an integer-valued float, the result is a floating-point approximation. If the number is a non-integral real number, the generalized factorial is used, as defined by the Euler Gamma function. Please note that computation of large factorials can be slow; using floating-point format will help since fewer digits must be maintained. The same is true of many of the commands in this section.

The k d (calc-double-factorial) [dfact] command computes the "double factorial" of an integer. For an even integer, this is the product of even integers from 2 to N. For an odd integer, this is the product of odd integers from 3 to N. If the argument is an integer-valued float, the result is a floating-point approximation. This function is undefined for negative even integers. The notation N!! is also recognized for double factorials.

The k c (calc-choose) [choose] command computes the binomial coefficient N-choose-M, where M is the number on the top of the stack and N is second-to-top. If both arguments are integers, the result is an exact integer. Otherwise, the result is a floating-point approximation. The binomial coefficient is defined for all real numbers by N! / M! (N-M)!.

The H k c (calc-perm) [perm] command computes the number-of-permutations function N! / (N-M)!.

The k b (calc-bernoulli-number) [bern] command computes a given Bernoulli number. The value at the top of the stack is a nonnegative integer n that specifies which Bernoulli number is desired. The H k b command computes a Bernoulli polynomial, taking n from the second-to-top position and x from the top of the stack. If x is a variable or formula the result is a polynomial in x; if x is a number the result is a number.

The k e (calc-euler-number) [euler] command similarly computes an Euler number, and H k e computes an Euler polynomial. Bernoulli and Euler numbers occur in the Taylor expansions of several functions.

The k s (calc-stirling-number) [stir1] command computes a Stirling number of the first kind , given two integers n and m on the stack. The H k s [stir2] command computes a Stirling number of the second kind . These are the number of m-cycle permutations of n objects, and the number of ways to partition n objects into m non-empty sets, respectively.

The k p (calc-prime-test) command checks if the integer on the top of the stack is prime. For integers less than eight million, the answer is always exact and reasonably fast. For larger integers, a probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P). The number is first checked against small prime factors (up to 13). Then, any number of iterations of the algorithm are performed. Each step either discovers that the number is non-prime, or substantially increases the certainty that the number is prime. After a few steps, the chance that a number was mistakenly described as prime will be less than one percent. (Indeed, this is a worst-case estimate of the probability; in practice even a single iteration is quite reliable.) After the k p command, the number will be reported as definitely prime or non-prime if possible, or otherwise "probably" prime with a certain probability of error.

The normal k p command performs one iteration of the primality test. Pressing k p repeatedly for the same integer will perform additional iterations. Also, k p with a numeric prefix performs the specified number of iterations. There is also an algebraic function `prime(n)' or `prime(n,iters)' which returns 1 if n is (probably) prime and 0 if not.

The k f (calc-prime-factors) [prfac] command attempts to decompose an integer into its prime factors. For numbers up to 25 million, the answer is exact although it may take some time. The result is a vector of the prime factors in increasing order. For larger inputs, prime factors above 5000 may not be found, in which case the last number in the vector will be an unfactored integer greater than 25 million (with a warning message). For negative integers, the first element of the list will be -1. For inputs -1, 0, and 1, the result is a list of the same number.

The k n (calc-next-prime) [nextprime] command finds the next prime above a given number. Essentially, it searches by calling calc-prime-test on successive integers until it finds one that passes the test. This is quite fast for integers less than eight million, but once the probabilistic test comes into play the search may be rather slow. Ordinarily this command stops for any prime that passes one iteration of the primality test. With a numeric prefix argument, a number must pass the specified number of iterations before the search stops. (This only matters when searching above eight million.) You can always use additional k p commands to increase your certainty that the number is indeed prime.

The I k n (calc-prev-prime) [prevprime] command analogously finds the next prime less than a given number.

The k t (calc-totient) [totient] command computes the Euler "totient" function , the number of integers less than n which are relatively prime to n.

The k m (calc-moebius) [moebius] command computes the Moebius "mu" function. If the input number is a product of k distinct factors, this is (-1)^k. If the input number has any duplicate factors (i.e., can be divided by the same prime more than once), the result is zero.


10.7 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. See section 12.7 Numerical Solutions.

