class

Exponential

extendsExponentialFamily

Exponential(rate: Tensor | float, validate_args: bool | None = None)

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Exponential distribution on $[0, \infty)$ .

Continuous distribution describing the waiting time between events of a Poisson process with rate $\lambda$ . It is the continuous analogue of the lucid.distributions.Geometric distribution and the unique continuous distribution with the memoryless property.

Parameters

rateTensor or float

Rate parameter

\lambda > 0

. The mean of the distribution is

1/\lambda

validate_argsbool= None

If True, validate parameter constraints at construction time.

Notes

Probability density on $x \geq 0$ :

p(x; \lambda) = \lambda e^{-\lambda x}

Moments:

\mathbb{E}[X] = \frac{1}{\lambda}, \qquad \mathrm{Var}[X] = \frac{1}{\lambda^2}, \qquad H[X] = 1 - \log \lambda

Memoryless property:

P(X > s + t \mid X > s) = P(X > t)

Special cases / relations:

$\mathrm{Exponential}(\lambda) = \mathrm{Gamma}(1, \lambda)$
The sum of $k$ IID $\mathrm{Exponential}(\lambda)$ variables is $\mathrm{Gamma}(k, \lambda)$ (Erlang).
$-\log U / \lambda$ with $U \sim \mathrm{Uniform}(0, 1)$ is the inverse-CDF sampler used by rsample.

Examples

>>> import lucid
>>> from lucid.distributions import Exponential
>>> d = Exponential(rate=2.0)
>>> d.mean  # 1/rate
Tensor(0.5)
>>> d.rsample((4,))
Tensor([...])
>>> d.log_prob(lucid.tensor(1.0))
Tensor(-1.3069)

Used by 2

Constructors

dunder

init

→None

__init__(rate: Tensor | float, validate_args: bool | None = None)

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Construct an Exponential distribution.

Parameters

rateTensor | float

Rate parameter

\lambda > 0

. The mean of the distribution is

1/\lambda

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Notes

The Exponential distribution with rate $\lambda$ has PDF:

p(x; \lambda) = \lambda e^{-\lambda x}, \quad x \geq 0

It is the continuous analogue of the Geometric distribution and describes the waiting time between events in a Poisson process.

Examples

>>> from lucid.distributions import Exponential
>>> d = Exponential(rate=2.0)
>>> d.mean  # 1/rate = 0.5
Tensor(0.5)

Properties

prop

mean

→Tensor

mean: Tensor

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Expected value of the Exponential distribution.

E[X] = \frac{1}{\lambda}

Returns

Tensor

Mean $1/\lambda$ , shape batch_shape.

Examples

>>> Exponential(rate=4.0).mean
Tensor(0.25)

prop

mode

→Tensor

mode: Tensor

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Mode of the Exponential distribution.

The Exponential distribution is monotonically decreasing on $[0, \infty)$ , so the mode is always zero regardless of the rate parameter.

\text{mode} = 0

Returns

Tensor

Zero tensor of shape batch_shape.

Examples

>>> Exponential(rate=5.0).mode
Tensor(0.0)

prop

variance

→Tensor

variance: Tensor

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Variance of the Exponential distribution.

\operatorname{Var}[X] = \frac{1}{\lambda^2}

Returns

Tensor

Variance $1/\lambda^2$ , shape batch_shape.

Examples

>>> Exponential(rate=2.0).variance
Tensor(0.25)

Instance methods

cdf

→Tensor

cdf(value: Tensor)

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Cumulative distribution function of the Exponential distribution.

F(x; \lambda) = 1 - e^{-\lambda x}, \quad x \geq 0

Parameters

valueTensor

Non-negative real values at which to evaluate the CDF.

Returns

Tensor

CDF values in $[0, 1]$ , shape batch_shape.

Examples

>>> Exponential(rate=1.0).cdf(lucid.tensor(1.0))  # 1 - e^-1 ≈ 0.632
Tensor(0.6321)

entropy

→Tensor

entropy()

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Shannon entropy of the Exponential distribution (in nats).

H(X) = 1 - \log \lambda

The entropy decreases as the rate increases (concentrated distributions have lower uncertainty).

Returns

Tensor

Entropy in nats, shape batch_shape.

Examples

>>> Exponential(rate=1.0).entropy()  # 1 - log(1) = 1.0
Tensor(1.0)

icdf

→Tensor

icdf(value: Tensor)

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Inverse CDF (quantile function) of the Exponential distribution.

F^{-1}(u; \lambda) = -\frac{\log(1 - u)}{\lambda}

Parameters

valueTensor

Probability values

u \in [0, 1)

Returns

Tensor

Quantiles in $[0, \infty)$ , shape batch_shape.

Examples

>>> Exponential(rate=1.0).icdf(lucid.tensor(0.5))  # log(2) ≈ 0.693
Tensor(0.6931)

log_prob

→Tensor

log_prob(value: Tensor)

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Log-density of value under the Exponential distribution.

\log p(x; \lambda) = \log \lambda - \lambda x

Parameters

valueTensor

Non-negative real values

x \geq 0

Returns

Tensor

Element-wise log-densities, shape batch_shape.

Examples

>>> d = Exponential(rate=1.0)
>>> d.log_prob(lucid.tensor(1.0))  # -1.0
Tensor(-1.0)

rsample

→Tensor

rsample(sample_shape: tuple[int, ...] = ())

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Reparameterised sample via the inverse-CDF method.

Uses the transform $X = -\log(1 - U) / \lambda$ where $U \sim \text{Uniform}(0, 1)$ . This is an exact reparameterisation — gradients flow through $\lambda$ .

X = -\frac{\log(1 - U)}{\lambda}

Subtracting from 1 (rather than directly using $U$ ) ensures the argument to $\log$ is bounded away from zero.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape dimensions for the sample batch. Default is ().

Returns

Tensor

Reparameterised samples of shape sample_shape + batch_shape.

Examples

>>> d = Exponential(rate=1.0)
>>> x = d.rsample((500,))
>>> x.mean()  # approximately 1.0

>>> import lucid >>> from lucid.distributions import Exponential >>> d = Exponential(rate=2.0) >>> d.mean # 1/rate Tensor(0.5) >>> d.rsample((4,)) Tensor([...]) >>> d.log_prob(lucid.tensor(1.0)) Tensor(-1.3069)

rsample

→Tensor

rsample(sample_shape: tuple[int, ...] = ())

source edit

Reparameterised sample via the inverse-CDF method.

Uses the transform $X = -\log(1 - U) / \lambda$ where $U \sim \text{Uniform}(0, 1)$ . This is an exact reparameterisation — gradients flow through $\lambda$ .

X = -\frac{\log(1 - U)}{\lambda}

Subtracting from 1 (rather than directly using $U$ ) ensures the argument to $\log$ is bounded away from zero.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape dimensions for the sample batch. Default is ().

Returns

Tensor

Reparameterised samples of shape sample_shape + batch_shape.

Examples

>>> d = Exponential(rate=1.0)
>>> x = d.rsample((500,))
>>> x.mean()  # approximately 1.0