# What are some examples of discrete variables

## Discrete random variable

In this chapter we look at what a discrete random variable is.

A function \ (X \) that

every result \ (\ omega \) of the result space \ (\ Omega \)

exactly one number \ (x \) of the set of real numbers \ (\ mathbb {R} \)

assigns is called **Random variable**.

*Short form:* \ (X: \ Omega \ rightarrow \ mathbb {R} \)

This definition can be illustrated very easily in a quantity diagram.

A random variable arranges

each \ (\ omega_i \) from \ (\ Omega \)

exactly one \ (x_i \) from \ (\ mathbb {R} \)

to.

Let yourself be of the word *Random variable* don't confuse! A random variable \ (X \) is not a number that comes out randomly in a random experiment, but a number *function*, which assigns a very precisely defined numerical value \ (x \) to every random result \ (\ omega \): \ (X: \ omega \ rightarrow x \).*Note:* The values \ (x \) assumed by a random variable \ (X \) are also called "realizations".

A random variable \ (X \) is called **discreet** designated,

if it only assumes a finitely many or a countably infinite number of values.

*Examples*

\ (X: = \ text {"Number of defective items in a sample"} \)

\ (\ Rightarrow \) finite set of values

\ (X: = \ text {"Number of throws until 6 appears for the first time"} \)

\ (\ Rightarrow \) infinite set of values, which however can be counted

Discrete random variables are usually created by one **Counting process**.

It follows that discrete random variables usually only take integer values.

**Examples of discrete random variables**

- the number of pips when throwing a dice
- the sum of the numbers when throwing multiple dice
- the number of times a die is thrown until 6 appears for the first time
- the number of times a coin has been thrown until \ (\ text {KOPF} \) is on top for the first time
- the number of defective items in a sample
- the number of products sold in a store in a day
- the number of claims that an insurance company has received in one year
- the win in a game of chance

### Discrete probability distribution

A **Probability distribution** indicates,

how the probabilities

distribute to the possible values of a random variable.

The probability distribution of a discrete random variable can be described by:

*example*

The random variable \ (X \) is the number of pips when throwing a symmetrical dice.

There are six possible realizations:

\ (x_1 = 1 \), \ (x_2 = 2 \), \ (x_3 = 3 \), \ (x_4 = 4 \), \ (x_5 = 5 \), \ (x_6 = 6 \)

All six realizations have the same probability:

\ (p_1 = p_2 = p_3 = p_4 = p_5 = p_6 = \ frac {1} {6} \)

**1.) Probability function**

\ [\ begin {equation *} f (x) = \ begin {cases} \ frac {1} {6} & \ text {for} x = 1 \ \ frac {1} {6} & \ text {for } x = 2 \ \ frac {1} {6} & \ text {for} x = 3 \ \ frac {1} {6} & \ text {for} x = 4 \ \ frac {1} { 6} & \ text {for} x = 5 \ \ frac {1} {6} & \ text {for} x = 6 \ 0 & \ text {otherwise} \ end {cases} \ end {equation *} \]

*Note:* \ (f (x) = P (X = x) \)

**2.) Distribution function**

\ [\ begin {equation *} F (x) = \ begin {cases} 0 & \ text {for} x <1 \ \ frac {1} {6} & \ text {for} 1 \ le x <2 \ \ frac {2} {6} & \ text {for} 2 \ le x <3 \ \ frac {3} {6} & \ text {for} 3 \ le x <4 \ \ frac {4 } {6} & \ text {for} 4 \ le x <5 \ \ frac {5} {6} & \ text {for} 5 \ le x <6 \ 1 & \ text {for} x \ ge 6 \ end {cases} \ end {equation *} \]

*Note:* \ (F (x) = P (X \ le x) \)

Both the *Probability function* as well as the *Distribution function* fully describe the probability distribution of a discrete random variable. A complete description of the distribution is often not necessary at all. In order to get a rough overview of a distribution, one looks at a few characteristic measures.

These include the expected value, the variance and the standard deviation.

### An overview of discrete random variables

Created by ... | Counting process |

Example: | Number of defective items in a sample |

Probability distribution | |

- probability function | |

- distribution function | |

Dimensions | |

- expected value | \ [\ mu_ {X} = \ mathrm {E} (X) = \ sum_i x_i \ cdot P (X = x_i) \] |

- variance | \ [\ sigma ^ 2_ {X} = \ mathrm {Var (X)} = \ sum_i (x_i - \ mu_ {X}) ^ 2 \ cdot P (X = x_i) \] |

- standard deviation | \ [\ sigma_ {X} = \ sqrt {\ mathrm {Var (x)}} \] |

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