Cauchy Distribution

stamox.distribution.pcauchy(q: Union[Float, ArrayLike], loc: Union[Float, ArrayLike] = 0.0, scale: Union[Float, ArrayLike] = 1.0, lower_tail: Bool = True, log_prob: Bool = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Calculates the cumulative denisty probability c function of the Cauchy distribution.

Parameters:

Name	Type	Description	Default
q	Union[Float, ArrayLike]	The value at which to evaluate the CDF.	required
loc	Union[Float, ArrayLike]	The location parameter of the Cauchy distribution. Defaults to 0.0.	0.0
scale	Union[Float, ArrayLike]	The scale parameter of the Cauchy distribution. Defaults to 1.0.	1.0
lower_tail	Bool	Whether to return the lower tail probability. Defaults to True.	True
log_prob	Bool	Whether to return the log probability. Defaults to False.	False
dtype	jnp.dtype	The dtype of the output. Defaults to jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	The cumulative density function of the Cauchy distribution.

Examples:

>>> pcauchy(1.0, loc=0.0, scale=1.0, lower_tail=True, log_prob=False)
Array(0.75, dtype=float32, weak_type=True)

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stamox.distribution.qcauchy(q: Union[Float, ArrayLike], loc: Union[Float, ArrayLike] = 0.0, scale: Union[Float, ArrayLike] = 1.0, lower_tail: Bool = True, log_prob: Bool = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Computes the quantile of the Cauchy distribution.

Parameters:

Name	Type	Description	Default
q	Union[float, array-like]	Quantiles to compute.	required
loc	Union[float, array-like]	Location parameter. Defaults to 0.0.	0.0
scale	Union[float, array-like]	Scale parameter. Defaults to 1.0.	1.0
lower_tail	bool	Whether to compute the lower tail. Defaults to True.	True
log_prob	bool	Whether to compute the log probability. Defaults to False.	False
dtype	jnp.dtype	The dtype of the output. Defaults to jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	The quantiles of the Cauchy distribution.

Examples:

>>> qcauchy(0.5, loc=1.0, scale=2.0)
Array([1.], dtype=float32, weak_type=True)

---

stamox.distribution.dcauchy(x: Union[Float, ArrayLike], loc: Union[Float, ArrayLike] = 0.0, scale: Union[Float, ArrayLike] = 1.0, lower_tail: Bool = True, log_prob: Bool = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Computes the pdf of the Cauchy distribution.

Parameters:

Name	Type	Description	Default
x	Union[Float, ArrayLike]	The input values.	required
loc	Union[Float, ArrayLike]	The location parameter. Defaults to 0.0.	0.0
scale	Union[Float, ArrayLike]	The scale parameter. Defaults to 1.0.	1.0
lower_tail	Bool	Whether to compute the lower tail. Defaults to True.	True
log_prob	Bool	Whether to compute the log probability. Defaults to False.	False
dtype	jnp.dtype	The dtype of the output. Defaults to jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	The pdf of the Cauchy distribution.

Examples:

>>> dcauchy(1.0, loc=0.0, scale=1.0)
Array([0.15915494], dtype=float32, weak_type=True)

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stamox.distribution.rcauchy(key: PRNGKeyArray, sample_shape: Optional[Sequence[int]] = None, loc: Union[Float, ArrayLike] = 0.0, scale: Union[Float, ArrayLike] = 1.0, lower_tail: Bool = True, log_prob: Bool = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Generates random samples from the Cauchy distribution.

Parameters:

Name	Type	Description	Default
key	PRNGKeyArray	A PRNGKey to use for generating the samples.	required
sample_shape	Optional[Sequence[int]]	The shape of the output array.	None
loc	Union[Float, ArrayLike]	The location parameter of the Cauchy distribution.	0.0
scale	Union[Float, ArrayLike]	The scale parameter of the Cauchy distribution.	1.0
lower_tail	Bool	Whether to return the lower tail probability.	True
log_prob	Bool	Whether to return the log probability.	False
dtype		The dtype of the output.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	An array of samples from the Cauchy distribution.

Examples:

>>> key = jax.random.PRNGKey(0)
>>> rcauchy(key, sample_shape=(2, 3), loc=0.0, scale=1.0)
Array([[ 0.23841971, -3.0880406 ,  0.9507532 ],
        [ 2.8963416 ,  0.31303588, -0.14792857]], dtype=float32)