Normal Distribution

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

Calculate the cumulative distribution function (CDF) of the normal distribution.

Parameters:

Name	Type	Description	Default
q	Union[Float, ArrayLike]	The quantiles to calculate the CDF at.	required
mean	Union[Float, ArrayLike]	The mean of the normal distribution. Defaults to 0.0.	0.0
sd	Union[Float, ArrayLike]	The standard deviation of the normal distribution. Defaults to 1.0.	1.0
lower_tail	bool	If True, calculate the probability that x is less than or equal to the given quantile(s). If False, calculate the probability that x is greater than the given quantile(s). Defaults to True.	True
log_prob	bool	If True, return the log of the CDF instead of the actual value. 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 CDF of the normal distribution evaluated at x.

Examples:

>>> pnorm(2.0)
Array(0.97724986, dtype=float32)

>>> pnorm([1.5, 2.0, 2.5], mean=2.0, sd=0.5, lower_tail=False)
Array([0.8413447 , 0.5       , 0.15865529], dtype=float32)

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stamox.distribution.qnorm(p: Union[Float, ArrayLike], mean: Union[Float, ArrayLike] = 0.0, sd: Union[Float, ArrayLike] = 1.0, lower_tail = True, log_prob = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Calculates the quantile function of the normal distribution for a given probability.

Parameters:

Name	Type	Description	Default
p	float or jnp.ndarray	Probability values.	required
mean	float or jnp.ndarray	Mean of the normal distribution. Default is 0.0.	0.0
sd	float or jnp.ndarray	Standard deviation of the normal distribution. Default is 1.0.	1.0
lower_tail	bool	If True, returns P(X ≤ x). If False, returns P(X > x). Default is True.	True
log_prob	bool	If True, returns the logarithm of the quantile function. Default is False.	False
dtype	jnp.dtype	The dtype of the output. Default is jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	The inverse cumulative density function of the normal distribution evaluated at q.

Examples:

>>> qnorm(0.5)
Array([0.], dtype=float32)
>>> qnorm([0.25, 0.75], mean=3, sd=2)
Array([1.6510204, 4.3489795], dtype=float32)

---

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

Probability density function (PDF) for Normal distribution.

Parameters:

Name	Type	Description	Default
x	Union[Float, ArrayLike]	The input value(s) at which to evaluate the PDF.	required
mean	Union[Float, ArrayLike]	The mean of the normal distribution. Defaults to 0.0.	0.0
sd	Union[Float, ArrayLike]	The standard deviation of the normal distribution. Defaults to 1.0.	1.0
lower_tail	bool	If True (default), returns the cumulative distribution function (CDF) from negative infinity up to x. Otherwise, returns the CDF from x to positive infinity.	True
log_prob	bool	If True, returns the log-probability instead of the probability.	False
dtype	jnp.dtype	The dtype of the output. Default is jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	The probability density function evaluated at point(s) x.

Examples:

>>> import jax.numpy as jnp
>>> x = jnp.array([0.5, 1.0, -1.5])
>>> dnorm(x)
Array([0.35206532, 0.24197075, 0.12951761], dtype=float32)

---

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

Generates random variables from a normal distribution.

Parameters:

Name	Type	Description	Default
key	PRNGKeyArray	A KeyArray object used to generate the random numbers.	required
sample_shape	Optional[Sequence[int]]	An optional tuple of integers specifying the shape of the	None
mean	Union[Float, ArrayLike]	The mean of the normal distribution. Defaults to 0.0.	0.0
sd	Union[Float, ArrayLike]	The standard deviation of the normal distribution. Defaults to 1.0.	1.0
lower_tail	Bool	If True (default), returns the cumulative distribution function (CDF) from negative infinity up to x. Otherwise, returns the CDF from x to positive infinity.	True
log_prob	Bool	If True, returns the log-probability instead of the probability.	False
dtype		The dtype of the output. Defaults to jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	Random samples from a normal distribution.

Examples:

>>> key = random.PRNGKey(0)
>>> rnorm(key, sample_shape=(3, 2))
Array([[ 0.18784384, -1.2833426 ],
        [ 0.6494181 ,  1.2490594 ],
        [ 0.24447003, -0.11744965]], dtype=float32)