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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