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# cumchi()

Usage:
 ```cumchi(x,df [,upper:T or lower:F]), x and df REAL, elements of df > 0 cumchi(x,df,lam [,upper:T or lower:F]), same x, df, lam REAL with 0 <= lam[i] < 1419.56542578676 ```

Keywords: probabilities
```cumchi(Val,df) computes P(x <= Val) where x is a chi-square random
variable with df degrees of freedom.  When Val is a vector, matrix or
array, the probabilities are computed for each element.  When Val and df
are both not scalars, they must be the same size and shape which will
also be the size and shape of the result.

The elements of df must be positive (fractional degrees of freedom are
allowed).

cumchi(Val, df, upper:T) and cumchi(Val, df, lower:F) do the same except
the upper tail probability P(x >= Val) is computed.  It is
mathematically the same as 1 - cumchi(Val, df), although a more accurate
value may be computed.

Example:
Cmd> 1 - cumchi(sum((obs-e)^2/e), nrows(obs) - 1) # P value

Cmd> cumchi(sum((obs-e)^2/e), nrows(obs) - 1, upper:T) # same

cumchi(Val,df,lam) computes P(x <= Val) where x is a non-central
chi-square random variable with df degrees of freedom and non-centrality
parameter lam.  Both cumchi(Val,df,lam,upper:T) and cumchi(Val,df,lam,
lower:F) compute P(X >= Val).  All non-scalar arguments must be the same
size and shape which will also be the size and shape of the result.  The
elements of Val must be non-negative and less than 1419.56542578676.

Power of 5% Chi-squared test of test of H_0: p[1] = p0[1], ..., p[k] =
p0[k] when p[j] = p1[j], j = 1,...,k based on a multinomial sample of
size n:
Cmd> cumchi(invchi(.05,k-1,upper:T),k-1,n*sum((p1-p0)^2/p0),upper:T)