# niklas@web: ~/Playground

## ðŸŽ° Playground TAG

This is a place where I’m playing around with different org features and the org publish function.

### âœ’ LaTeX

\(\LaTeX\) support: \(\varphi = \sum_{i} a_i^2 + b_i^2\).

#### Markov Chains

##### Definition

A sequence of random variables \(X_1, \dots X_T\) which fulfills the **Markov property**:
\[ P(X_t \mid X_1, \dots, X_{t-1}) = P(X_t \mid X_{t-1})\]
where

- Time indices \(t\) are discrete
- Assume that the random variables \(X_t\) are discrete

Joint distribution: \[ P(X_t = i_1, \dots, X_T = i_T) = P(X_1 = i_1) \prod_{t=1}^{T-1} P(X_{t+1} = i_{t+1} \mid X_t = i_t) \]

##### General Case

\[ P(X_1 = i) = \pi_i \\ P(X_{t+1} = j \mid X_t = i) = A_{ij}^{(t+1)} \]

where \(\pi \in \mathbb{R}^K\) is a **prior probability** on the initial state and \(A^{(t)} \in \mathbb{R}^{K \times K}\) are the **transition matrices**.

Thus, we have a joint probability of \[ P(X_1=i_1, \dots, X_T=i_T) = \pi_{i1} \times A_{i_1,i_2}^{(2)} \times \dots \times A^{(T)}_{i_{T-1}, i_T} \]

##### Stationary Case

To simplify, assume a **time-homogeneous** or **stationary** Markov Chain:
\[ P(X_1 = i) = \pi_i \\ P(X_{t+1} = j \mid X_t = i) = A_{ij} \]

The tranisiton matrix \(A^{(t)} = A\) does not depend on \(t\).

### ðŸ”— Links

Links can be written in plain text: http://www.niklasbuehler.com, or formatted: Home.

### ðŸ’» Code

Bash with text output:

echo "A"

A

Python with image output:

import matplotlib import matplotlib.pyplot as plt fig=plt.figure(figsize=(3,2)) plt.plot([1,3,2]) fig.tight_layout() fname = "res/img/myfig.png" plt.savefig(fname) fname # return this to org-mode

### ðŸ§® Automatic Spreadsheet formatting

Name | Grade | ECTS | Weighted Grade |
---|---|---|---|

Spanish B1.1 | 1.0 | 3 | 3. |

MLRG | 1.3 | 6 | 7.8 |

TN | 2.3 | 5 | 11.5 |

Total | 1.5333333 | 14 | 1.5928571 |