RandomState

class brainpy.math.random.RandomState(seed_or_key=None, seed=None, ready_to_trace=True)

RandomState that track the random generator state.

bernoulli(p, size=None, key=None)

Sample Bernoulli random values with given shape and mean.

Parameters:

• p (float, array_like, optional) – A float or array of floats for the mean of the random variables. Must be broadcast-compatible with shape and the values should be within [0, 1]. Default 0.5.

• size (optional, tuple of int, int) – A tuple of nonnegative integers representing the result shape. Must be broadcast-compatible with p.shape. The default (None) produces a result shape equal to p.shape.

Returns:

out – A random array with boolean dtype and shape given by shape if shape is not None, or else p.shape.

Return type:

array_like

beta(a, b, size=None, key=None)

Draw samples from a Beta distribution.

The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function

\[f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1},\]

where the normalization, B, is the beta function,

\[B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt.\]

It is often seen in Bayesian inference and order statistics.

Parameters:

• a (float or array_like of floats) – Alpha, positive (>0).

• b (float or array_like of floats) – Beta, positive (>0).

• size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a and b are both scalars. Otherwise, np.broadcast(a, b).size samples are drawn.

Returns:

out – Drawn samples from the parameterized beta distribution.

Return type:

ndarray or scalar

See also

random.Generator.beta

which should be used for new code.

categorical(logits, axis=-1, size=None, key=None)

Sample random values from categorical distributions.

Parameters:

• logits – Unnormalized log probabilities of the categorical distribution(s) to sample from, so that softmax(logits, axis) gives the corresponding probabilities.

• axis (int) – Axis along which logits belong to the same categorical distribution.

• shape – Optional, a tuple of nonnegative integers representing the result shape. Must be broadcast-compatible with np.delete(logits.shape, axis). The default (None) produces a result shape equal to np.delete(logits.shape, axis).

• key – a PRNG key used as the random key.

Returns:

A random array with int dtype and shape given by shape if shape is not None, or else np.delete(logits.shape, axis).

choice(a, size=None, replace=True, p=None, key=None)

Generates a random sample from a given 1-D array

Parameters:

• a (1-D array-like or int) – If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if it were np.arange(a)

• size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

• replace (boolean, optional) – Whether the sample is with or without replacement. Default is True, meaning that a value of a can be selected multiple times.

• p (1-D array-like, optional) – The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in a.

Returns:

samples – The generated random samples

Return type:

single item or ndarray

Raises:

ValueError – If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size

See also

randint, shuffle, permutation

Generator.choice

which should be used in new code

Notes

Setting user-specified probabilities through p uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of p is 1 / len(a).

Sampling random rows from a 2-D array is not possible with this function, but is possible with Generator.choice through its axis keyword.

Examples

Generate a uniform random sample from np.arange(5) of size 3:

>>> import brainpy.math as bm
>>> bm.random.choice(5, 3)
array([0, 3, 4]) # random
>>> #This is equivalent to brainpy.math.random.randint(0,5,3)

Generate a non-uniform random sample from np.arange(5) of size 3:

>>> bm.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0]) # random

Generate a uniform random sample from np.arange(5) of size 3 without replacement:

>>> bm.random.choice(5, 3, replace=False)
array([3,1,0]) # random
>>> #This is equivalent to brainpy.math.random.permutation(np.arange(5))[:3]

Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:

>>> bm.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0]) # random

Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance:

>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> bm.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
      dtype='<U11')

clone()

Clone the variable.

loggamma(a, size=None, key=None)

Sample log-gamma random values.

Parameters:

• a (float, array_like) – A float or array of floats broadcast-compatible with shape representing the parameter of the distribution.

• size (optional, int, tuple of int) – A tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with a. The default (None) produces a result shape equal to a.shape.

Returns:

out – The sampled results.

Return type:

array_like

maxwell(size=None, key=None)

Sample from a one sided Maxwell distribution.

The scipy counterpart is scipy.stats.maxwell.

Parameters:

• key – a PRNG key.

• size – The shape of the returned samples.

• dtype – The type used for samples.

Returns:

A jnp.array of samples, of shape shape.

orthogonal(n, size=None, key=None)

Sample uniformly from the orthogonal group O(n).

Parameters:

• n (int) – An integer indicating the resulting dimension.

• size (optional, int, tuple of int) – The batch dimensions of the result.

Returns:

out – The sampled results.

Return type:

Array

permutation(x, axis=0, independent=False, key=None)

Randomly permute a sequence, or return a permuted range.

If x is a multi-dimensional array, it is only shuffled along its first index.

Parameters:

x (int or array_like) – If x is an integer, randomly permute np.arange(x). If x is an array, make a copy and shuffle the elements randomly.

Returns:

out – Permuted sequence or array range.

Return type:

ndarray

See also

random.Generator.permutation

which should be used for new code.

Examples

>>> import brainpy.math as bm
>>> bm.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random

>>> bm.random.permutation([1, 4, 9, 12, 15])
array([15,  1,  9,  4, 12]) # random

>>> arr = np.arange(9).reshape((3, 3))
>>> bm.random.permutation(arr)
array([[6, 7, 8], # random
       [0, 1, 2],
       [3, 4, 5]])

rand(*dn, key=None)

Random values in a given shape.

Note

This is a convenience function for users porting code from Matlab, and wraps random_sample. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Create an array of the given shape and populate it with random samples from a uniform distribution over [0, 1).

Parameters:

• d0 (int, optional) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

• d1 (int, optional) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

• ... (int, optional) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

• dn (int, optional) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

Returns:

out – Random values.

Return type:

ndarray, shape (d0, d1, ..., dn)

See also

random

Examples

>>> brainpy.math.random.rand(3,2)
array([[ 0.14022471,  0.96360618],  #random
       [ 0.37601032,  0.25528411],  #random
       [ 0.49313049,  0.94909878]]) #random

rand_like(input, *, dtype=None, key=None)

Returns a tensor with the same size as input that is filled with random numbers from a uniform distribution on the interval [0, 1).

Parameters:

• input – the size of input will determine size of the output tensor.

• dtype – the desired data type of returned Tensor. Default: if None, defaults to the dtype of input.

• key – the seed or key for the random.

Returns:

The random data.

randint(low, high=None, size=None, dtype=<class 'int'>, key=None)

Return random integers from low (inclusive) to high (exclusive).

Return random integers from the “discrete uniform” distribution of the specified dtype in the “half-open” interval [low, high). If high is None (the default), then results are from [0, low).

Parameters:

• low (int or array-like of ints) – Lowest (signed) integers to be drawn from the distribution (unless high=None, in which case this parameter is one above the highest such integer).

• high (int or array-like of ints, optional) – If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None). If array-like, must contain integer values

• size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

• dtype (dtype, optional) – Desired dtype of the result. Byteorder must be native. The default value is int.

Returns:

out – size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.

Return type:

int or ndarray of ints

See also

random_integers

similar to randint, only for the closed interval [low, high], and 1 is the lowest value if high is omitted.

Generator.integers

which should be used for new code.

Examples

>>> import brainpy.math as bm
>>> bm.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random
>>> bm.random.randint(1, size=10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

Generate a 2 x 4 array of ints between 0 and 4, inclusive:

>>> bm.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1], # random
       [3, 2, 2, 0]])

Generate a 1 x 3 array with 3 different upper bounds

>>> bm.random.randint(1, [3, 5, 10])
array([2, 2, 9]) # random

Generate a 1 by 3 array with 3 different lower bounds

>>> bm.random.randint([1, 5, 7], 10)
array([9, 8, 7]) # random

Generate a 2 by 4 array using broadcasting with dtype of uint8

>>> bm.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)
array([[ 8,  6,  9,  7], # random
       [ 1, 16,  9, 12]], dtype=uint8)

randint_like(input, low=0, high=None, *, dtype=None, key=None)

Similar to randint_like in torch.

Returns a tensor with the same shape as Tensor input filled with random integers generated uniformly between low (inclusive) and high (exclusive).

Parameters:

• input – the size of input will determine size of the output tensor.

• low – Lowest integer to be drawn from the distribution. Default: 0.

• high – One above the highest integer to be drawn from the distribution.

• dtype – the desired data type of returned Tensor. Default: if None, defaults to the dtype of input.

• key – the seed or key for the random.

Returns:

The random data.

randn(*dn, key=None)

Return a sample (or samples) from the “standard normal” distribution.

Note

This is a convenience function for users porting code from Matlab, and wraps standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.

Note

New code should use the standard_normal method of a default_rng() instance instead; please see the random-quick-start.

If positive int_like arguments are provided, randn generates an array of shape (d0, d1, ..., dn), filled with random floats sampled from a univariate “normal” (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided.

Parameters:

• d0 (int, optional) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

• d1 (int, optional) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

• ... (int, optional) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

• dn (int, optional) – The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

Returns:

Z – A (d0, d1, ..., dn)-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied.

Return type:

ndarray or float

See also

standard_normal

Similar, but takes a tuple as its argument.

normal

Also accepts mu and sigma arguments.

random.Generator.standard_normal

which should be used for new code.

Notes

For random samples from \(N(\mu, \sigma^2)\), use:

sigma * bm.random.randn(...) + mu

Examples

>>> import brainpy.math as bm
>>> bm.random.randn()
2.1923875335537315  # random

Two-by-four array of samples from N(3, 6.25):

>>> 3 + 2.5 * bm.random.randn(2, 4)
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

randn_like(input, *, dtype=None, key=None)

Returns a tensor with the same size as input that is filled with random numbers from a normal distribution with mean 0 and variance 1.

Parameters:

• input – the size of input will determine size of the output tensor.

• dtype – the desired data type of returned Tensor. Default: if None, defaults to the dtype of input.

• key – the seed or key for the random.

Returns:

The random data.

random(size=None, key=None)

Return random floats in the half-open interval [0.0, 1.0). Alias for random_sample to ease forward-porting to the new random API.

random_integers(low, high=None, size=None, key=None)

Random integers of type np.int_ between low and high, inclusive.

Return random integers of type np.int_ from the “discrete uniform” distribution in the closed interval [low, high]. If high is None (the default), then results are from [1, low]. The np.int_ type translates to the C long integer type and its precision is platform dependent.

Parameters:

• low (int) – Lowest (signed) integer to be drawn from the distribution (unless high=None, in which case this parameter is the highest such integer).

• high (int, optional) – If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None).

• size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

Returns:

out – size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.

Return type:

int or ndarray of ints

See also

randint

Similar to random_integers, only for the half-open interval [low, high), and 0 is the lowest value if high is omitted.

Notes

To sample from N evenly spaced floating-point numbers between a and b, use:

a + (b - a) * (bm.random.random_integers(N) - 1) / (N - 1.)

Examples

>>> import brainpy.math as bm
>>> bm.random.random_integers(5)
4 # random
>>> type(bm.random.random_integers(5))
<class 'numpy.int64'>
>>> bm.random.random_integers(5, size=(3,2))
array([[5, 4], # random
       [3, 3],
       [4, 5]])

Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (i.e., from the set \({0, 5/8, 10/8, 15/8, 20/8}\)):

>>> 2.5 * (bm.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625,  1.25 ,  0.625,  0.625,  2.5  ]) # random

Roll two six sided dice 1000 times and sum the results:

>>> d1 = bm.random.random_integers(1, 6, 1000)
>>> d2 = bm.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2

Display results as a histogram:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, density=True)
>>> plt.show()

random_sample(size=None, key=None)

Return random floats in the half-open interval [0.0, 1.0).

Results are from the “continuous uniform” distribution over the stated interval. To sample \(Unif[a, b), b > a\) multiply the output of random_sample by (b-a) and add a:

(b - a) * random_sample() + a

Note

New code should use the random method of a default_rng() instance instead; please see the random-quick-start.

Parameters:

size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

Returns:

out – Array of random floats of shape size (unless size=None, in which case a single float is returned).

Return type:

float or ndarray of floats

See also

Generator.random

which should be used for new code.

Examples

>>> import brainpy.math as bm
>>> bm.random.random_sample()
0.47108547995356098 # random
>>> type(bm.random.random_sample())
<class 'float'>
>>> bm.random.random_sample((5,))
array([ 0.30220482,  0.86820401,  0.1654503 ,  0.11659149,  0.54323428]) # random

Three-by-two array of random numbers from [-5, 0):

>>> 5 * bm.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984], # random
       [-2.99091858, -0.79479508],
       [-1.23204345, -1.75224494]])

ranf(size=None, key=None)

This is an alias of random_sample. See random_sample for the complete

documentation.

sample(size=None, key=None)

This is an alias of random_sample. See random_sample for the complete

documentation.

seed(seed_or_key=None, seed=None)

Sets a new random seed.

Parameters:

• seed_or_key (int, ArrayType, optional) –

  It can be an integer for initial seed of the random number generator, or it can be a JAX’s PRNKey, which is an array with two elements and uint32 dtype.

  New in version 2.2.3.4.

• seed (int, ArrayType, optional) –

  Same as seed_or_key.

  Deprecated since version 2.2.3.4: Will be removed since version 2.4.

shuffle(x, axis=0, key=None)

Modify a sequence in-place by shuffling its contents.

This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.

Parameters:

x (ndarray or MutableSequence) – The array, list or mutable sequence to be shuffled.

Return type:

None

See also

random.Generator.shuffle

which should be used for new code.

Examples

>>> import brainpy.math as bm
>>> arr = np.arange(10)
>>> bm.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8] # random

Multi-dimensional arrays are only shuffled along the first axis:

>>> arr = np.arange(9).reshape((3, 3))
>>> bm.random.shuffle(arr)
>>> arr
array([[3, 4, 5], # random
       [6, 7, 8],
       [0, 1, 2]])

split_key()

Create a new seed from the current seed.

split_keys(n)

Create multiple seeds from the current seed. This is used internally by pmap and vmap to ensure that random numbers are different in parallel threads.

Parameters:

n (int) – The number of seeds to generate.

t(df, size=None, key=None)

Sample Student’s t random values.

Parameters:

• df (float, array_like) – A float or array of floats broadcast-compatible with shape representing the parameter of the distribution.

• size (optional, int, tuple of int) – A tuple of non-negative integers specifying the result shape. Must be broadcast-compatible with df. The default (None) produces a result shape equal to df.shape.

Returns:

out – The sampled value.

Return type:

array_like

truncated_normal(lower, upper, size=None, scale=None, key=None)

Sample truncated standard normal random values with given shape and dtype.

Parameters:

• lower (float, ndarray) – A float or array of floats representing the lower bound for truncation. Must be broadcast-compatible with upper.

• upper (float, ndarray) – A float or array of floats representing the upper bound for truncation. Must be broadcast-compatible with lower.

• size (optional, list of int, tuple of int) – A tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with lower and upper. The default (None) produces a result shape by broadcasting lower and upper.

• scale (float, ndarray) – Standard deviation (spread or “width”) of the distribution. Must be non-negative.

Returns:

out – A random array with the specified dtype and shape given by shape if shape is not None, or else by broadcasting lower and upper. Returns values in the open interval (lower, upper).

Return type:

Array

weibull(a, size=None, key=None)

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.

\[X = (-ln(U))^{1/a}\]

Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter \(\lambda\) is just \(X = \lambda(-ln(U))^{1/a}\).

Note

New code should use the weibull method of a default_rng() instance instead; please see the random-quick-start.

Parameters:

• a (float or array_like of floats) – Shape parameter of the distribution. Must be nonnegative.

• size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.

Returns:

out – Drawn samples from the parameterized Weibull distribution.

Return type:

ndarray or scalar

See also

scipy.stats.weibull_max, scipy.stats.weibull_min, scipy.stats.genextreme, gumbel

random.Generator.weibull

which should be used for new code.

Notes

The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions.

The probability density for the Weibull distribution is

\[p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},\]

where \(a\) is the shape and \(\lambda\) the scale.

The function has its peak (the mode) at \(\lambda(\frac{a-1}{a})^{1/a}\).

When a = 1, the Weibull distribution reduces to the exponential distribution.

References

[1]

Waloddi Weibull, Royal Technical University, Stockholm, 1939 “A Statistical Theory Of The Strength Of Materials”, Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.

[2]

Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951.

[3]

Wikipedia, “Weibull distribution”, https://en.wikipedia.org/wiki/Weibull_distribution

Examples

Draw samples from the distribution:

>>> a = 5. # shape
>>> s = brainpy.math.random.weibull(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)

>>> count, bins, ignored = plt.hist(brainpy.math.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()

weibull_min(a, scale=None, size=None, key=None)

Sample from a Weibull minimum distribution.

Parameters:

• a (float, array_like) – The concentration parameter of the distribution.

• scale (float, array_like) – The scale parameter of the distribution.

• size (optional, int, tuple of int) – The shape added to the parameters loc and scale broadcastable shape.

Returns:

out – The sampling results.

Return type:

array_like

zipf(a, size=None, key=None)

Draw samples from a Zipf distribution.

Samples are drawn from a Zipf distribution with specified parameter a > 1.

The Zipf distribution (also known as the zeta distribution) is a discrete probability distribution that satisfies Zipf’s law: the frequency of an item is inversely proportional to its rank in a frequency table.

Note

New code should use the zipf method of a default_rng() instance instead; please see the random-quick-start.

Parameters:

• a (float or array_like of floats) – Distribution parameter. Must be greater than 1.

• size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.

Returns:

out – Drawn samples from the parameterized Zipf distribution.

Return type:

ndarray or scalar

See also

scipy.stats.zipf

probability density function, distribution, or cumulative density function, etc.

random.Generator.zipf

which should be used for new code.

Notes

The probability density for the Zipf distribution is

\[p(k) = \frac{k^{-a}}{\zeta(a)},\]

for integers \(k \geq 1\), where \(\zeta\) is the Riemann Zeta function.

It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table.

References

[1]

Zipf, G. K., “Selected Studies of the Principle of Relative Frequency in Language,” Cambridge, MA: Harvard Univ. Press, 1932.

Examples

Draw samples from the distribution:

>>> a = 4.0
>>> n = 20000
>>> s = brainpy.math.random.zipf(a, n)

Display the histogram of the samples, along with the expected histogram based on the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import zeta  

bincount provides a fast histogram for small integers.

>>> count = np.bincount(s)
>>> k = np.arange(1, s.max() + 1)

>>> plt.bar(k, count[1:], alpha=0.5, label='sample count')
>>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5,
...          label='expected count')   
>>> plt.semilogy()
>>> plt.grid(alpha=0.4)
>>> plt.legend()
>>> plt.title(f'Zipf sample, a={a}, size={n}')
>>> plt.show()