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scipy.stats.Normal.

# cdf

Normal.cdf(x, y=None, /, *, method=None)

  Cumulative distribution function

  The cumulative distribution function (“CDF”), denoted \(F(x)\), is the probability the random variable \(X\) will assume a value less than or equal to \(x\):

  \[F(x) = P(X ≤ x)\]

  A two-argument variant of this function is also defined as the probability the random variable \(X\) will assume a value between \(x\) and \(y\).

  \[F(x, y) = P(x ≤ X ≤ y)\]

  `cdf` accepts x for \(x\) and y for \(y\).

  Parameters:
    x, y array_like

      The arguments of the CDF. x is required; y is optional.

    method{None, ‘formula’, ‘logexp’, ‘complement’, ‘quadrature’, ‘subtraction’}

      The strategy used to evaluate the CDF. By default (`None`), the one-argument form of the function chooses between the following options, listed in order of precedence.

      - `'formula'`: use a formula for the CDF itself

      - `'logexp'`: evaluate the log-CDF and exponentiate

      - `'complement'`: evaluate the CCDF and take the complement

      - `'quadrature'`: numerically integrate the PDF (or, in the discrete case, sum the PMF)

      In place of `'complement'`, the two-argument form accepts:

      - `'subtraction'`: compute the CDF at each argument and take the difference.

      Not all method options are available for all distributions. If the selected method is not available, a `NotImplementedError` will be raised.

  Returns:
    out array

      The CDF evaluated at the provided argument(s).

  See also

  `logcdf`
  `ccdf`

  Notes

  Suppose a continuous probability distribution has support \([l, r]\). The CDF \(F(x)\) is related to the probability density function \(f(x)\) by:

  \[F(x) = \int_l^x f(u) du\]

  The two argument version is:

  \[F(x, y) = \int_x^y f(u) du = F(y) - F(x)\]

  The CDF evaluates to its minimum value of \(0\) for \(x ≤ l\) and its maximum value of \(1\) for \(x ≥ r\).

  Suppose a discrete probability distribution has support \([l, r]\). The CDF \(F(x)\) is related to the probability mass function \(f(x)\) by:

  \[F(x) = \sum_{u=l}^{\lfloor x \rfloor} f(u)\]

  The CDF evaluates to its minimum value of \(0\) for \(x < l\) and its maximum value of \(1\) for \(x ≥ r\).

  The CDF is also known simply as the “distribution function”.

  References

  [1]

  Cumulative distribution function, Wikipedia, https://en.wikipedia.org/wiki/Cumulative_distribution_function

  Examples

  Instantiate a distribution with the desired parameters:

      >>> from scipy import stats
      >>> X = stats.Uniform(a=-0.5, b=0.5)

  Evaluate the CDF at the desired argument:

      >>> X.cdf(0.25)
      0.75

  Evaluate the cumulative probability between two arguments:

      >>> X.cdf(-0.25, 0.25) == X.cdf(0.25) - X.cdf(-0.25)
      True