class_docstr: Student's t-distribution.
  
  This distribution has parameters: degree of freedom `df`, location `loc`,
  and `scale`.
  
  #### Mathematical details
  
  The probability density function (pdf) is,
  
  ```none
  pdf(x; df, mu, sigma) = (1 + y**2 / df)**(-0.5 (df + 1)) / Z
  where,
  y = (x - mu) / sigma
  Z = abs(sigma) sqrt(df pi) Gamma(0.5 df) / Gamma(0.5 (df + 1))
  ```
  
  where:
  * `loc = mu`,
  * `scale = sigma`, and,
  * `Z` is the normalization constant, and,
  * `Gamma` is the [gamma function](
    https://en.wikipedia.org/wiki/Gamma_function).
  
  The StudentT distribution is a member of the [location-scale family](
  https://en.wikipedia.org/wiki/Location-scale_family), i.e., it can be
  constructed as,
  
  ```none
  X ~ StudentT(df, loc=0, scale=1)
  Y = loc + scale * X
  ```
  
  Notice that `scale` has semantics more similar to standard deviation than
  variance. However it is not actually the std. deviation; the Student's
  t-distribution std. dev. is `scale sqrt(df / (df - 2))` when `df > 2`.
  
  Samples of this distribution are reparameterized (pathwise differentiable).
  The derivatives are computed using the approach described in
  (Figurnov et al., 2018).
  
  #### Examples
  
  Examples of initialization of one or a batch of distributions.
  
  ```python
  import tensorflow_probability as tfp
  tfd = tfp.distributions
  
  # Define a single scalar Student t distribution.
  single_dist = tfd.StudentT(df=3)
  
  # Evaluate the pdf at 1, returning a scalar Tensor.
  single_dist.prob(1.)
  
  # Define a batch of two scalar valued Student t's.
  # The first has degrees of freedom 2, mean 1, and scale 11.
  # The second 3, 2 and 22.
  multi_dist = tfd.StudentT(df=[2, 3], loc=[1, 2.], scale=[11, 22.])
  
  # Evaluate the pdf of the first distribution on 0, and the second on 1.5,
  # returning a length two tensor.
  multi_dist.prob([0, 1.5])
  
  # Get 3 samples, returning a 3 x 2 tensor.
  multi_dist.sample(3)
  ```
  
  Arguments are broadcast when possible.
  
  ```python
  # Define a batch of two Student's t distributions.
  # Both have df 2 and mean 1, but different scales.
  dist = tfd.StudentT(df=2, loc=1, scale=[11, 22.])
  
  # Evaluate the pdf of both distributions on the same point, 3.0,
  # returning a length 2 tensor.
  dist.prob(3.0)
  ```
  
  Compute the gradients of samples w.r.t. the parameters:
  
  ```python
  df = tf.constant(2.0)
  loc = tf.constant(2.0)
  scale = tf.constant(11.0)
  dist = tfd.StudentT(df=df, loc=loc, scale=scale)
  samples = dist.sample(5)  # Shape [5]
  loss = tf.reduce_mean(tf.square(samples))  # Arbitrary loss function
  # Unbiased stochastic gradients of the loss function
  grads = tf.gradients(loss, [df, loc, scale])
  ```
  
  References:
    Implicit Reparameterization Gradients:
      [Figurnov et al., 2018]
      (http://papers.nips.cc/paper/7326-implicit-reparameterization-gradients)
      ([pdf](http://papers.nips.cc/paper/7326-implicit-reparameterization-gradients.pdf))
code: def _event_shape(self):
      return tensor_shape.TensorShape([])
  
imports: import numpy
  from tensorflow.python.framework import constant_op
  from tensorflow.python.framework import dtypes
  from tensorflow.python.framework import ops
  from tensorflow.python.framework import tensor_shape
  from tensorflow.python.ops import array_ops
  from tensorflow.python.ops import check_ops
  from tensorflow.python.ops import control_flow_ops
  from tensorflow.python.ops import math_ops
  from tensorflow.python.ops import nn
  from tensorflow.python.ops import random_ops
  from tensorflow.python.ops import special_math_ops
  from tensorflow.python.ops.distributions import distribution
  from tensorflow.python.ops.distributions import util
  from tensorflow.python.util import deprecation
  from tensorflow.python.util.tf_export import tf_export
label_desc: Use image node_id 9 for calling the StudentT obj's underlying member method code with example usage: obj._event_shape() and returns: tensor_shape