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cumbin(x,N,P [,upper:T or lower:F]), x, N and P REAL, elements of N
  integers > 0, elements of P between 0 and 1

Keywords: probabilities
cumbin(Val,N,P) computes the probabilities that a binomial random
variable with N trials at probability P would be <= elements of the
vector, matrix or array Val.  When Val is not integral, the result is
the same as cumbin(floor(Val),N,P).

Any of Val, N, and P that are not scalars (single numbers) must be
vectors, matrices, or arrays with the same size and shape which will
also be the size and shape of the result.

The elements of N must be positive integers less than 100,000 and the
elements of P must be between zero and one.

cumbin(Val,N,P,upper:T) and cumbin(Val,N,P,lower:F) compute the
probability that the binomial random variable is >= elements of Val.
This is mathematically the same as 1 - cumbin(ceiling(Val - 1),N,P), not
1 - cumbin(Val,N,P)

Note that when Val is an integer, P(x = Val) is included in both
cumbin(Val,N,P) and cumbin(Val,N,P,upper:T).

If x_obs is an observed number of successes in a sequence of n
independent trials with constant p = P(success), you can use cumbin() to
compute P-values for a test of H_0: p = p_0 as follows:
    H_a      P-value
  p > p_0    cumbin(x_obs,n,p_0,upper:T)
  p < p_0    cumbin(x_obs,n,p_0)
  p != p_0   2*min(cumbin(x_obs,n,p_0),cumbin(x_obs,n,p_0,upper:T))

  Cmd> 1 - cumbin(7,13,.25) # P(x > 7) with n = 13, p = .25
  (1)   0.0056493

  Cmd> cumbin(8,13,.25,upper:T) # or cumbin(8,13,.25,lower:F), same
  (1)   0.0056493

See also cumpoi().  See invbeta:"binomical_confidence_interval" for
information on computing a confidence interval for p based on a binomial
random variable

Gary Oehlert 2003-01-15