def fdtrc(a, b, x):
"""Returns right tail of F distribution, x to infinity.
See Cephes docs for details.
"""
if min(a, b) < 1:
raise ValueError, "F a and b (degrees of freedom) must both be >= 1."
if x < 0:
raise ValueError, "F distribution value of f must be >= 0."
w = float(b) / (b + a * x)
return betai(0.5 * b, 0.5 * a, w)
def fdtr(a, b, x):
"""Returns left tail of F distribution, 0 to x.
See Cephes docs for details.
"""
if min(a, b) < 1:
raise ValueError, "F a and b (degrees of freedom) must both be >= 1."
if x < 0:
raise ValueError, "F distribution value of f must be >= 0."
w = a * x
w /= float(b + w)
return betai(0.5 * a, 0.5 * b, w)
def stdtr(k, t):
"""Student's t distribution, -infinity to t.
See Cephes docs for details.
"""
if k <= 0:
raise ValueError, 'stdtr: df must be > 0.'
if t == 0:
return 0.5
if t < -2:
rk = k
z = rk / (rk + t * t)
return 0.5 * betai(0.5 * rk, 0.5, z)
#compute integral from -t to + t
if t < 0:
x = -t
else:
x = t
rk = k #degrees of freedom
z = 1 + (x * x)/rk
#test if k is odd or even
if (k & 1) != 0:
#odd k
xsqk = x/sqrt(rk)
p = atan(xsqk)
if k > 1:
f = 1
tz = 1
j = 3
while (j <= (k-2)) and ((tz/f) > MACHEP):
tz *= (j-1)/(z*j)
f += tz
j += 2
p += f * xsqk/z
p *= 2/PI
else:
#even k
f = 1
tz = 1
j = 2
while (j <= (k-2)) and ((tz/f) > MACHEP):
tz *= (j-1)/(z*j)
f += tz
j += 2
p = f * x/sqrt(z*rk)
#common exit
if t < 0:
p = -p #note destruction of relative accuracy
p = 0.5 + 0.5 * p
return p
def bdtr(k, n, p):
"""Binomial distribution, 0 through k.
Uses formula bdtr(k, n, p) = betai(n-k, k+1, 1-p)
See Cephes docs for details.
"""
p = fix_rounding_error(p)
if (p < 0) or (p > 1):
raise ValueError, "Binomial p must be between 0 and 1."
if (k < 0) or (n < k):
raise ValueError, "Binomial k must be between 0 and n."
if k == n:
return 1
dn = n - k
if k == 0:
return pow(1-p, dn)
else:
return betai(dn, k+1, 1-p)
def bdtrc(k, n, p):
"""Complement of binomial distribution, k+1 through n.
Uses formula bdtrc(k, n, p) = betai(k+1, n-k, p)
See Cephes docs for details.
"""
p = fix_rounding_error(p)
if (p < 0) or (p > 1):
raise ValueError, "Binomial p must be between 0 and 1."
if (k < 0) or (n < k):
raise ValueError, "Binomial k must be between 0 and n."
if k == n:
return 0
dn = n - k
if k == 0:
if p < .01:
dk = -expm1(dn * log1p(-p))
else:
dk = 1 - pow(1.0-p, dn)
else:
dk = k + 1
dk = betai(dk, dn, p)
return dk