# dit/dit

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```.. py:module:: dit.npscalardist

```

## Numpy-based ScalarDistribution

ScalarDistributions are used to represent distributions over real numbers, for example a six-sided die or the number of heads when flipping 100 coins.

### Playing with ScalarDistributions

First we will enable two optional features: printing fractions by default, and using :func:`__str__` as :func:`__repr__`. Be careful using either of these options, they can incur significant performance hits on some distributions.

```.. ipython::

In [1]: dit.ditParams['print.exact'] = dit.ditParams['repr.print'] = True

```

We next construct a six-sided die:

```.. ipython::

In [2]: from dit.example_dists import uniform

In [3]: d6 = uniform(1, 7)

@doctest
In [4]: d6
Class:    ScalarDistribution
Alphabet: (1, 2, 3, 4, 5, 6)
Base:     linear

x   p(x)
1   1/6
2   1/6
3   1/6
4   1/6
5   1/6
6   1/6

```

We can perform standard mathematical operations with scalars, such as adding, subtracting from or by, multiplying, taking the modulo, or testing inequalities.

```.. ipython::

@doctest
In [5]: d6 + 3
Class:    ScalarDistribution
Alphabet: (4, 5, 6, 7, 8, 9)
Base:     linear

x   p(x)
4   1/6
5   1/6
6   1/6
7   1/6
8   1/6
9   1/6

@doctest
In [6]: d6 - 1
Class:    ScalarDistribution
Alphabet: (0, 1, 2, 3, 4, 5)
Base:     linear

x   p(x)
0   1/6
1   1/6
2   1/6
3   1/6
4   1/6
5   1/6

@doctest
In [7]: 10 - d6
Class:    ScalarDistribution
Alphabet: (4, 5, 6, 7, 8, 9)
Base:     linear

x   p(x)
4   1/6
5   1/6
6   1/6
7   1/6
8   1/6
9   1/6

@doctest
In [8]: 2 * d6
Class:    ScalarDistribution
Alphabet: (2, 4, 6, 8, 10, 12)
Base:     linear

x    p(x)
2    1/6
4    1/6
6    1/6
8    1/6
10   1/6
12   1/6

@doctest
In [9]: d6 % 2
Class:    ScalarDistribution
Alphabet: (0, 1)
Base:     linear

x   p(x)
0   1/2
1   1/2

@doctest
In [10]: (d6 % 2).is_approx_equal(d6 <= 3)
Out[10]: True

```

Furthermore, we can perform such operations with two distributions:

```.. ipython::

@doctest
In [11]: d6 + d6
Class:    ScalarDistribution
Alphabet: (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
Base:     linear

x    p(x)
2    1/36
3    1/18
4    1/12
5    1/9
6    5/36
7    1/6
8    5/36
9    1/9
10   1/12
11   1/18
12   1/36

@doctest
In [12]: (d6 + d6) % 4
Class:    ScalarDistribution
Alphabet: (0, 1, 2, 3)
Base:     linear

x   p(x)
0   1/4
1   2/9
2   1/4
3   5/18

@doctest
In [13]: d6 // d6
Class:    ScalarDistribution
Alphabet: (0, 1, 2, 3, 4, 5, 6)
Base:     linear

x   p(x)
0   5/12
1   1/3
2   1/9
3   1/18
4   1/36
5   1/36
6   1/36

@doctest
In [14]:  d6 % (d6 % 2 + 1)
Class:    ScalarDistribution
Alphabet: (0, 1)
Base:     linear

x   p(x)
0   3/4
1   1/4

```

There are also statistical functions which can be applied to :class:`~dit.ScalarDistributions`:

```.. ipython::

In [15]: from dit.algorithms.stats import *

@doctest float
In [16]: median(d6+d6)
Out[16]: 7.0

In [17]: from dit.example_dists import binomial

In [18]: d = binomial(10, 1/3)

@doctest
In [19]: d
Class:    ScalarDistribution
Alphabet: (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
Base:     linear

x    p(x)
0    409/23585
1    4302/49615
2    1280/6561
3    5120/19683
4    4480/19683
5    896/6561
6    1120/19683
7    320/19683
8    20/6561
9    9/26572
10   1/59046

@doctest float
In [20]: mean(d)
Out[20]: 3.3333333333333335

@doctest float
In [21]: median(d)
Out[21]: 3.0

@doctest float
In [22]: standard_deviation(d)
Out[22]: 1.4907119849998596

```

### API

```.. automethod:: ScalarDistribution.__init__
```