Advanced probability functions

Analytica User GuideOther FunctionsAdvanced probability functions

The following functions are not actual probability distributions, but they are useful for various probabilistic analyses, including building other probability distributions. You can find them in the Advanced math library from the Definition menu.

BetaFn(a, b): The beta function, defined as:

[math]\displaystyle{ BetaFn(a, b) = \int\limits_{0}^{1}x^{a -1}(1-x)^{b-1}dx }[/math]

BetaI(x, a, b): The incomplete beta function, defined as:

[math]\displaystyle{ BetaI(x, a, b) = \frac{1}{Beta(a,b)}\int\limits_{0}^{X}x^{a -1}(1-x)^{b-1}dx }[/math]

The incomplete beta function is equal to the cumulative probability of the beta distribution at x. It is useful in a number of mathematical and statistical applications.

The cumulative binomial distribution, defined as the probability that an event with probability p occurs k or more times in n trials, is given by:

[math]\displaystyle{ Pr = BetaI(p, k, n - k + 1) }[/math]

The student’s distribution with n degrees of freedom, used to test whether two observed distributions have the same mean, is readily available from the beta distribution as:

[math]\displaystyle{ Student(x|n) = 1 - BetaI(n/(n + x^2), n/2, 1/2) }[/math]

The F-distribution, used to test whether two observed samples with n<sub1>and n₂degrees of freedom have the same variance, is readily obtained from BetaI as:

[math]\displaystyle{ F(x, n_1, n_2) = BetaI(_2 / (n_{1}x + n_2)) }[/math]

BetaIaInv(p, x, b): The inverse of the incomplete beta function with respect to the second parameter, a. Returns the value a such that BetaI(x, a, b) = p.

BetaIInv(p, a, b): The inverse of the incomplete beta function with respect to the first parameter, x. Returns the value x such that BetaI(x, a, b) = p.

Combinations(k, n): “n choose k.” The number of unique ways that k items can be chosen from a set of n elements (without replacement and ignoring the order).

Combinations(2, 4) &rarr 6

They are: {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}

Permutations(k, n): The number of possible permutations of k items taken from a bucket of n items.

Permutations(2, 4) → 12

They are: {1,2}, {1,3}, {1,4}, {2,1}, {2,3}, {2,4}, {3,1}, {3,2}, {3,4}, {4,1}, {4,2}, {4,3}

CumNormal(x, mean, stddev): Returns the cumulative probability:

[math]\displaystyle{ p = Pr[x ≤ X] }[/math]

for a normal distribution with a given mean and standard deviation. mean and stddev are optional and default to mean = 0, stddev = 1.

CumNormal(1) - CumNormal(-1) → .683

That is, 68.3% of the area under a normal distribution is contained within one standard deviation of the mean.

CumNormalInv(p, m, s): The inverse cumulative probability function for a normal distribution with mean m and standard deviation s. This is also called the Probit function. Returns the value X where:

[math]\displaystyle{ p = Pr[x ≤ X] }[/math]

mean and stddev are optional and default to mean = 0, stddev = 1.

Erf(x): The error function, defined as:

See Also

Financial library functionsAdvanced probability functionsExpressing Uncertainty