How to do it...

To create a Markov chain for the study - and - sleep process and conduct some analysis on it, perform the following steps:

1. Import the library and define the transition matrix:

>>> import torch
>>> T = torch.tensor([[0.4, 0.6],
...                   [0.8, 0.2]])

1. Calculate the transition probability after k steps. Here, we use k = 2, 5, 10, 15, and 20 as examples:

>>> T_2 = torch.matrix_power(T, 2)
>>> T_5 = torch.matrix_power(T, 5)
>>> T_10 = torch.matrix_power(T, 10)
>>> T_15 = torch.matrix_power(T, 15)
>>> T_20 = torch.matrix_power(T, 20)

1. Define the initial distribution of two states:

>>> v = torch.tensor([[0.7, 0.3]])

1. Calculate the state distribution after k = 1, 2, 5, 10, 15, and 20 steps:

>>> v_1 = torch.mm(v, T)
>>> v_2 = torch.mm(v, T_2)
>>> v_5 = torch.mm(v, T_5)
>>> v_10 = torch.mm(v, T_10)
>>> v_15 = torch.mm(v, T_15)
>>> v_20 = torch.mm(v, T_20)