# tf.compat.v1.distributions.Distribution

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|---|
| [![](https://www.tensorflow.org/images/GitHub-Mark-32px.png) View source on GitHub](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L272-L1316) |

A generic probability distribution base class.

    tf.compat.v1.distributions.Distribution(
        dtype,
        reparameterization_type,
        validate_args,
        allow_nan_stats,
        parameters=None,
        graph_parents=None,
        name=None
    )

`Distribution` is a base class for constructing and organizing properties
(e.g., mean, variance) of random variables (e.g, Bernoulli, Gaussian).

#### Subclassing

Subclasses are expected to implement a leading-underscore version of the
same-named function. The argument signature should be identical except for
the omission of `name="..."`. For example, to enable `log_prob(value,
name="log_prob")` a subclass should implement `_log_prob(value)`.

Subclasses can append to public-level docstrings by providing
docstrings for their method specializations. For example:

    @util.AppendDocstring("Some other details.")
    def _log_prob(self, value):
      ...

would add the string "Some other details." to the `log_prob` function
docstring. This is implemented as a simple decorator to avoid python
linter complaining about missing Args/Returns/Raises sections in the
partial docstrings.

#### Broadcasting, batching, and shapes

All distributions support batches of independent distributions of that type.
The batch shape is determined by broadcasting together the parameters.

The shape of arguments to `__init__`, `cdf`, `log_cdf`, `prob`, and
`log_prob` reflect this broadcasting, as does the return value of `sample` and
`sample_n`.

`sample_n_shape = [n] + batch_shape + event_shape`, where `sample_n_shape` is
the shape of the `Tensor` returned from `sample_n`, `n` is the number of
samples, `batch_shape` defines how many independent distributions there are,
and `event_shape` defines the shape of samples from each of those independent
distributions. Samples are independent along the `batch_shape` dimensions, but
not necessarily so along the `event_shape` dimensions (depending on the
particulars of the underlying distribution).

Using the `Uniform` distribution as an example:

    minval = 3.0
    maxval = [[4.0, 6.0],
              [10.0, 12.0]]

    # Broadcasting:
    # This instance represents 4 Uniform distributions. Each has a lower bound at
    # 3.0 as the `minval` parameter was broadcasted to match `maxval`'s shape.
    u = Uniform(minval, maxval)

    # `event_shape` is `TensorShape([])`.
    event_shape = u.event_shape
    # `event_shape_t` is a `Tensor` which will evaluate to [].
    event_shape_t = u.event_shape_tensor()

    # Sampling returns a sample per distribution. `samples` has shape
    # [5, 2, 2], which is [n] + batch_shape + event_shape, where n=5,
    # batch_shape=[2, 2], and event_shape=[].
    samples = u.sample_n(5)

    # The broadcasting holds across methods. Here we use `cdf` as an example. The
    # same holds for `log_cdf` and the likelihood functions.

    # `cum_prob` has shape [2, 2] as the `value` argument was broadcasted to the
    # shape of the `Uniform` instance.
    cum_prob_broadcast = u.cdf(4.0)

    # `cum_prob`'s shape is [2, 2], one per distribution. No broadcasting
    # occurred.
    cum_prob_per_dist = u.cdf([[4.0, 5.0],
                               [6.0, 7.0]])

    # INVALID as the `value` argument is not broadcastable to the distribution's
    # shape.
    cum_prob_invalid = u.cdf([4.0, 5.0, 6.0])

#### Shapes

There are three important concepts associated with TensorFlow Distributions
shapes:

- Event shape describes the shape of a single draw from the distribution; it may be dependent across dimensions. For scalar distributions, the event shape is `[]`. For a 5-dimensional MultivariateNormal, the event shape is `[5]`.
- Batch shape describes independent, not identically distributed draws, aka a "collection" or "bunch" of distributions.
- Sample shape describes independent, identically distributed draws of batches from the distribution family.

The event shape and the batch shape are properties of a Distribution object,
whereas the sample shape is associated with a specific call to `sample` or
`log_prob`.

For detailed usage examples of TensorFlow Distributions shapes, see
[this tutorial](https://github.com/tensorflow/probability/blob/master/tensorflow_probability/examples/jupyter_notebooks/Understanding_TensorFlow_Distributions_Shapes.ipynb)

#### Parameter values leading to undefined statistics or distributions.

Some distributions do not have well-defined statistics for all initialization
parameter values. For example, the beta distribution is parameterized by
positive real numbers `concentration1` and `concentration0`, and does not have
well-defined mode if `concentration1 < 1` or `concentration0 < 1`.

The user is given the option of raising an exception or returning `NaN`.

    a = tf.exp(tf.matmul(logits, weights_a))
    b = tf.exp(tf.matmul(logits, weights_b))

    # Will raise exception if ANY batch member has a < 1 or b < 1.
    dist = distributions.beta(a, b, allow_nan_stats=False)
    mode = dist.mode().eval()

    # Will return NaN for batch members with either a < 1 or b < 1.
    dist = distributions.beta(a, b, allow_nan_stats=True)  # Default behavior
    mode = dist.mode().eval()

In all cases, an exception is raised if *invalid* parameters are passed, e.g.

    # Will raise an exception if any Op is run.
    negative_a = -1.0 * a  # beta distribution by definition has a > 0.
    dist = distributions.beta(negative_a, b, allow_nan_stats=True)
    dist.mean().eval()

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| ## Args ||
|---|---|
| `dtype` | The type of the event samples. `None` implies no type-enforcement. |
| `reparameterization_type` | Instance of `ReparameterizationType`. If [`distributions.FULLY_REPARAMETERIZED`](https://www.tensorflow.org/probability/oryx/api_docs/python/oryx/distributions#FULLY_REPARAMETERIZED), this `Distribution` can be reparameterized in terms of some standard distribution with a function whose Jacobian is constant for the support of the standard distribution. If [`distributions.NOT_REPARAMETERIZED`](https://www.tensorflow.org/probability/oryx/api_docs/python/oryx/distributions#NOT_REPARAMETERIZED), then no such reparameterization is available. |
| `validate_args` | Python `bool`, default `False`. When `True` distribution parameters are checked for validity despite possibly degrading runtime performance. When `False` invalid inputs may silently render incorrect outputs. |
| `allow_nan_stats` | Python `bool`, default `True`. When `True`, statistics (e.g., mean, mode, variance) use the value "`NaN`" to indicate the result is undefined. When `False`, an exception is raised if one or more of the statistic's batch members are undefined. |
| `parameters` | Python `dict` of parameters used to instantiate this `Distribution`. |
| `graph_parents` | Python `list` of graph prerequisites of this `Distribution`. |
| `name` | Python `str` name prefixed to Ops created by this class. Default: subclass name. |

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| ## Raises ||
|---|---|
| `ValueError` | if any member of graph_parents is `None` or not a `Tensor`. |

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| ## Attributes ||
|---|---|
| `allow_nan_stats` | Python `bool` describing behavior when a stat is undefined. <br /> Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E\[(X - mean)\*\*2\] is also undefined. |
| `batch_shape` | Shape of a single sample from a single event index as a `TensorShape`. <br /> May be partially defined or unknown. The batch dimensions are indexes into independent, non-identical parameterizations of this distribution. |
| `dtype` | The `DType` of `Tensor`s handled by this `Distribution`. |
| `event_shape` | Shape of a single sample from a single batch as a `TensorShape`. <br /> May be partially defined or unknown. |
| `name` | Name prepended to all ops created by this `Distribution`. |
| `parameters` | Dictionary of parameters used to instantiate this `Distribution`. |
| `reparameterization_type` | Describes how samples from the distribution are reparameterized. <br /> Currently this is one of the static instances [`distributions.FULLY_REPARAMETERIZED`](https://www.tensorflow.org/probability/oryx/api_docs/python/oryx/distributions#FULLY_REPARAMETERIZED) or [`distributions.NOT_REPARAMETERIZED`](https://www.tensorflow.org/probability/oryx/api_docs/python/oryx/distributions#NOT_REPARAMETERIZED). |
| `validate_args` | Python `bool` indicating possibly expensive checks are enabled. |

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## Methods

### `batch_shape_tensor`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L630-L647)

    batch_shape_tensor(
        name='batch_shape_tensor'
    )

Shape of a single sample from a single event index as a 1-D `Tensor`.

The batch dimensions are indexes into independent, non-identical
parameterizations of this distribution.

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| Args ||
|---|---|
| `name` | name to give to the op |

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| Returns ||
|---|---|
| `batch_shape` | `Tensor`. |

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### `cdf`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L874-L891)

    cdf(
        value, name='cdf'
    )

Cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

    cdf(x) := P[X <= x]

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `cdf` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `copy`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L608-L624)

    copy(
        **override_parameters_kwargs
    )

Creates a deep copy of the distribution.

> [!NOTE]
> **Note:** the copy distribution may continue to depend on the original initialization arguments.

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| Args ||
|---|---|
| `**override_parameters_kwargs` | String/value dictionary of initialization arguments to override with new values. |

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| Returns ||
|---|---|
| `distribution` | A new instance of `type(self)` initialized from the union of self.parameters and override_parameters_kwargs, i.e., `dict(self.parameters, **override_parameters_kwargs)`. |

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### `covariance`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L1087-L1124)

    covariance(
        name='covariance'
    )

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-`k`, vector-valued distribution, it is calculated
as,

    Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]

where `Cov` is a (batch of) `k x k` matrix, `0 <= (i, j) < k`, and `E`
denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g.,
matrix-valued, Wishart), `Covariance` shall return a (batch of) matrices
under some vectorization of the events, i.e.,

    Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]

where `Cov` is a (batch of) `k' x k'` matrices,
`0 <= (i, j) < k' = reduce_prod(event_shape)`, and `Vec` is some function
mapping indices of this distribution's event dimensions to indices of a
length-`k'` vector.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `covariance` | Floating-point `Tensor` with shape `[B1, ..., Bn, k', k']` where the first `n` dimensions are batch coordinates and `k' = reduce_prod(self.event_shape)`. |

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### `cross_entropy`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L1139-L1162)

    cross_entropy(
        other, name='cross_entropy'
    )

Computes the (Shannon) cross entropy.

Denote this distribution (`self`) by `P` and the `other` distribution by
`Q`. Assuming `P, Q` are absolutely continuous with respect to
one another and permit densities `p(x) dr(x)` and `q(x) dr(x)`, (Shanon)
cross entropy is defined as:

    H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)

where `F` denotes the support of the random variable `X ~ P`.

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| Args ||
|---|---|
| `other` | [`tfp.distributions.Distribution`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Distribution) instance. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `cross_entropy` | `self.dtype` `Tensor` with shape `[B1, ..., Bn]` representing `n` different calculations of (Shanon) cross entropy. |

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### `entropy`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L975-L978)

    entropy(
        name='entropy'
    )

Shannon entropy in nats.

### `event_shape_tensor`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L670-L684)

    event_shape_tensor(
        name='event_shape_tensor'
    )

Shape of a single sample from a single batch as a 1-D int32 `Tensor`.

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| Args ||
|---|---|
| `name` | name to give to the op |

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| Returns ||
|---|---|
| `event_shape` | `Tensor`. |

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### `is_scalar_batch`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L714-L726)

    is_scalar_batch(
        name='is_scalar_batch'
    )

Indicates that `batch_shape == []`.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `is_scalar_batch` | `bool` scalar `Tensor`. |

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### `is_scalar_event`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L700-L712)

    is_scalar_event(
        name='is_scalar_event'
    )

Indicates that `event_shape == []`.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `is_scalar_event` | `bool` scalar `Tensor`. |

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### `kl_divergence`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L1168-L1194)

    kl_divergence(
        other, name='kl_divergence'
    )

Computes the Kullback--Leibler divergence.

Denote this distribution (`self`) by `p` and the `other` distribution by
`q`. Assuming `p, q` are absolutely continuous with respect to reference
measure `r`, the KL divergence is defined as:

    KL[p, q] = E_p[log(p(X)/q(X))]
             = -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
             = H[p, q] - H[p]

where `F` denotes the support of the random variable `X ~ p`, `H[., .]`
denotes (Shanon) cross entropy, and `H[.]` denotes (Shanon) entropy.

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<br />

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| Args ||
|---|---|
| `other` | [`tfp.distributions.Distribution`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Distribution) instance. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `kl_divergence` | `self.dtype` `Tensor` with shape `[B1, ..., Bn]` representing `n` different calculations of the Kullback-Leibler divergence. |

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### `log_cdf`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L835-L856)

    log_cdf(
        value, name='log_cdf'
    )

Log cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

    log_cdf(x) := Log[ P[X <= x] ]

Often, a numerical approximation can be used for `log_cdf(x)` that yields
a more accurate answer than simply taking the logarithm of the `cdf` when
`x << -1`.

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `logcdf` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `log_prob`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L777-L788)

    log_prob(
        value, name='log_prob'
    )

Log probability density/mass function.

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `log_prob` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `log_survival_function`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L910-L932)

    log_survival_function(
        value, name='log_survival_function'
    )

Log survival function.

Given random variable `X`, the survival function is defined:

    log_survival_function(x) = Log[ P[X > x] ]
                             = Log[ 1 - P[X <= x] ]
                             = Log[ 1 - cdf(x) ]

Typically, different numerical approximations can be used for the log
survival function, which are more accurate than `1 - cdf(x)` when `x >> 1`.

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. ||

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### `mean`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L984-L987)

    mean(
        name='mean'
    )

Mean.

### `mode`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L1130-L1133)

    mode(
        name='mode'
    )

Mode.

### `param_shapes`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L490-L509)

    @classmethod
    param_shapes(
        sample_shape, name='DistributionParamShapes'
    )

Shapes of parameters given the desired shape of a call to `sample()`.

This is a class method that describes what key/value arguments are required
to instantiate the given `Distribution` so that a particular shape is
returned for that instance's call to `sample()`.

Subclasses should override class method `_param_shapes`.

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| Args ||
|---|---|
| `sample_shape` | `Tensor` or python list/tuple. Desired shape of a call to `sample()`. |
| `name` | name to prepend ops with. |

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| Returns ||
|---|---|
| `dict` of parameter name to `Tensor` shapes. ||

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### `param_static_shapes`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L511-L548)

    @classmethod
    param_static_shapes(
        sample_shape
    )

param_shapes with static (i.e. `TensorShape`) shapes.

This is a class method that describes what key/value arguments are required
to instantiate the given `Distribution` so that a particular shape is
returned for that instance's call to `sample()`. Assumes that the sample's
shape is known statically.

Subclasses should override class method `_param_shapes` to return
constant-valued tensors when constant values are fed.

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| Args ||
|---|---|
| `sample_shape` | `TensorShape` or python list/tuple. Desired shape of a call to `sample()`. |

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| Returns ||
|---|---|
| `dict` of parameter name to `TensorShape`. ||

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| Raises ||
|---|---|
| `ValueError` | if `sample_shape` is a `TensorShape` and is not fully defined. |

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### `prob`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L806-L817)

    prob(
        value, name='prob'
    )

Probability density/mass function.

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `prob` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `quantile`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L999-L1016)

    quantile(
        value, name='quantile'
    )

Quantile function. Aka "inverse cdf" or "percent point function".

Given random variable `X` and `p in [0, 1]`, the `quantile` is:

    quantile(p) := x such that P[X <= x] == p

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `quantile` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `sample`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L745-L759)

    sample(
        sample_shape=(), seed=None, name='sample'
    )

Generate samples of the specified shape.

Note that a call to `sample()` without arguments will generate a single
sample.

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| Args ||
|---|---|
| `sample_shape` | 0D or 1D `int32` `Tensor`. Shape of the generated samples. |
| `seed` | Python integer seed for RNG |
| `name` | name to give to the op. |

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| Returns ||
|---|---|
| `samples` | a `Tensor` with prepended dimensions `sample_shape`. |

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### `stddev`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L1054-L1081)

    stddev(
        name='stddev'
    )

Standard deviation.

Standard deviation is defined as,

    stddev = E[(X - E[X])**2]**0.5

where `X` is the random variable associated with this distribution, `E`
denotes expectation, and `stddev.shape = batch_shape + event_shape`.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `stddev` | Floating-point `Tensor` with shape identical to `batch_shape + event_shape`, i.e., the same shape as `self.mean()`. |

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### `survival_function`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L950-L969)

    survival_function(
        value, name='survival_function'
    )

Survival function.

Given random variable `X`, the survival function is defined:

    survival_function(x) = P[X > x]
                         = 1 - P[X <= x]
                         = 1 - cdf(x).

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. ||

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### `variance`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.14.0/tensorflow/python/ops/distributions/distribution.py#L1022-L1048)

    variance(
        name='variance'
    )

Variance.

Variance is defined as,

    Var = E[(X - E[X])**2]

where `X` is the random variable associated with this distribution, `E`
denotes expectation, and `Var.shape = batch_shape + event_shape`.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `variance` | Floating-point `Tensor` with shape identical to `batch_shape + event_shape`, i.e., the same shape as `self.mean()`. |

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