What are possible random variables

Random variables and distributions

In this article, we'll explain everything about random variables and distributions. We discuss the following sub-topics in detail:

Our Mathe-Abi'21 learning booklets Explanations ✔ Examples ✔ Free learning videos ✔

New!

What is a random variable?

A random variable ($ X $) is a function that assigns real numbers to the results of a random experiment:

\ begin {align *}
X: \ Omega \ rightarrow \ mathbb {R}, \ quad X: \ omega \ rightarrow X (\ omega) = x
\ end {align *}

example 1

Roll the dice twice with the random variable "$ X $ = total"

$
\ begin {array} {c | c | c | c | c | c | c}
e_i & (1 | 1) & (1 | 2) & (2 | 1) & (2 | 2) & \ dots & (6 | 6) \ \ hline
X (e_i) = x_i & 2 & 3 & 3 & 4 & \ dots & 12
\ end {array} $

Note: The realization (rolled number) can be achieved by different throw combinations. So one can achieve the realization “4” with the throws (2 | 2), (1 | 3) and (3 | 1). The probability 3/36 results from the number of natural events.


Example 2

A coin is tossed twice with the random variable "$ X $ = number of number throws"

What are the options? What does the result space look like? Knowing this, we can create the following table:

$
\ begin {array} {c | c | c | c}
e_i & (K | K) & (Z | K), (K | Z) & (Z | Z) \ \ hline
X (e_i) = x_i & 0 & 1 & 2 \ \ hline
P (x_i) & 1/4 & 1/2 & 1/4
\ end {array}
$

Daniel explains again what a random variable is.

Random size and probability distribution, basics with example | Math by Daniel Jung

Math Abi'21 study booklet incl. Exercise collection

New!

Discrete random variables

A random variable is called discreet, if it at last or countable infinite values $ X_1 $, $ X_2 $, $ X_3 $, $ \ dots $, $ X_n $.

A random variable $ (X) $, which only has a finite or countably infinite number of values, is always discrete.

Examples: number of pips when rolling the dice, tossing a coin

Carrier of a discrete random variable

  • The carrier $ T_X $ of a discrete $ ZV $ $ X $ is the set of all values ​​that $ X $ assumes with a positive probability.
  • Usually the carrier of a discrete random variable is a subset of the natural numbers $ (1,2,3, \ dots, n) $.
  • The carrier $ T_X $ can also be understood as an event space.
  • Important: First of all, all carriers of the random variables must always be determined in order to then determine the probability.

Examples of discrete random variables

  • Number of pips when rolling the dice once (event space $ \ Omega = \ {1,2,3,4,5,6 \}) $.
  • Number of times a normal deck of cards was drawn without replacing until the 7 of diamonds is drawn ($ \ Omega = \ {1,2,3, \ dots, 32 \} $).

Our Mathe-Abi'21 learning booklets Explanations ✔ Examples ✔ Free learning videos ✔

New!

Probability function of a discrete random variable

A probability distribution indicates how the probabilities are distributed among the possible values ​​of a random variable and is only defined for discrete random variables.

Definition:

\ begin {align *}
f: \ mathbb {R} \ rightarrow [0; 1], \ quad f: x \ rightarrow f (x) = P (X = x) = p
\ end {align *}

$ P (X = x) $ indicates the probability that the ZV $ X $ assumes the value $ x $. It might seem complicated at first, but let's look at a small example:

A normal die is thrown. As already described, the carriers of the random variables must first be determined, which can also be understood as an event space. So the event space is

\ begin {align *}
\ Omega = \ {1,2,3,4,5,6 \}.
\ end {align *}

The probabilities are then calculated using the Laplace probability, analogous to the previous probabilities.


Our Mathe-Abi'21 learning booklets Explanations ✔ Examples ✔ Free learning videos ✔

New!

Distribution function of a discrete random variable

The distribution function is a tool for describing a discrete (or continuous) probability distribution. A function $ F $ that assigns exactly one probability $ P (X \ leq x) $ to each $ x $ of a random variable $ X $ is called a distribution function.

\ begin {align *}
F: \ mathbb {R} \ rightarrow [0,1], \ quad
F: x \ rightarrow \ F (x) = P (X≤x)
\ end {align *}

Ah yes, and what does that mean? Interpretation:

The distribution function measures the probability that the random variable $ X $ takes on at most the value $ x $: $ F (X) = P (X \ leq x) $ = “Probability that $ X $ is less than or equal to a certain value $ x $ Has."

Properties:

  • The distribution function of a discrete random variable is a step function.
  • $ F (x) $ is defined for each $ x $ and takes values ​​from 0 to 1.
  • Used in hypothesis tests! Signal word: At most.

Note:

\ begin {align *}
P (X \ geq x) = 1-P (X x) = 1-P (X \ leq x)
\ end {align *}

example

A coin is tossed twice with the random variable $ X $ = number of "number throws"!

$
\ begin {array} {c | c | c | c}
e_i & (K | K) & (Z | K), (K | Z) & (Z | Z) \ \ hline
X (e_i) = x_i & 0 & 1 & 2 \ \ hline
P (x_i) & 1/4 & 1/2 & 1/4 \ \ hline
F (X) & 0.25 & 0.75 & 1 \
\ end {array}
$

1) How high is the probability that a “number” will be thrown at least once in the above random experiment?

Basically, we express the operator at least as $ \ geq $ and write what is sought from the task. It follows:

\ begin {align *}
P (X \ geq 1)
\ end {align *}

In the second step it can be seen from the task that the carriers of the random variables are $ \ Omega = \ {0,1,2 \} $.

At this point you have to ask yourself what is really greater than or equal to one. From this it follows that only $ P (X = 1) $ and $ P (X = 2) $ are really greater than or equal to one and the probabilities for both random variables are known.

\ begin {align *}
P (X \ geq 1) = P (X = 1) + P (X = 2) = 0.5 +0.25 = 0.75
\ end {align *}

Alternatively, you can proceed as follows: Ask yourself what is really less than one. Only $ P (X = 0) $ is really less than one, and the probability of this is known. It follows

\ begin {align *}
P (X \ geq 1) = 1- P (X <1) = 1 - P (X = 0) = 1 - 0.25 = 0.75
\ end {align *}

2) What is the probability that a “number” will be thrown at most once in a random experiment?

Basically, we express the operator at most as $ \ leq $ and write out what is sought from the task. It follows:

\ begin {align *}
P (X \ leq 1)
\ end {align *}

At this point you have to ask yourself what is really less than or equal to one. From this it follows that only $ P (X = 0) $ and $ P (X = 1) $ are really less than or equal to one and the probabilities for both random variables are known.

\ begin {align *}
P (X \ leq 1) = P (X = 0) + P (X = 1) = 0.25 +0.5 = 0.75
\ end {align *}

> You can also make use of the distribution function when answering, since the distribution function measures the probability that the random variable $ X $ at most ($ \ leq $) takes on the value $ x $! It follows

\ begin {align *}
P (X \ leq 1) = F (X = 1) = 0.75.
\ end {align *}

Take a closer look at Daniel's video tutorial on the topic of distribution functions.

Distribution function, cumulative, stochastics, probability theory, math by Daniel Jung

Math Abi'21 study booklet incl. Exercise collection

New!

Distribution parameter of a discrete random variable

Distribution parameters are quantities that characterize certain aspects of a distribution, such as the position, scatter or skewness of a distribution.

Important parameters are:

Expected value (location parameter):

  • The expected value is the center of gravity of the distribution and describes the number that the random variable assumes on average.
  • The expected value $ E (X) $ is also often referred to as $ \ mu $.

\ begin {align *}
\ mu = E (X) = \ sum_ {i = 1} ^ k x_i \ cdot \ underbrace {P (X = x_i)} _ {= p_i} = x_1 \ cdot p_1 + x_2 \ cdot p_2 + \ dots + x_k \ cdot p_k
\ end {align *}

Variance (dispersion parameter):

  • Variance describes the spread of a random variable and does not depend on chance.
  • The variance of the random variable $ X $ is the expected value of the squared deviation from its expected value.
  • Often $ \ sigma ^ 2 $ is written instead of $ V (X) $.

\ begin {align *}
\ sigma ^ 2 = V (X) = \ sum_ {i = 1} ^ k (x_i- \ mu) ^ 2 \ cdot p_i
\ end {align *}

  • The shift theorem $ \ sigma ^ 2 = \ sum_ {i = 1} ^ k x_i ^ 2 \ cdot p_i - \ mu ^ 2 $ usually simplifies the calculation of the variance.

Standard deviation (variance parameter):

  • The standard deviation is the positive square root of the variance and indicates the spread of the values ​​around the mean.
  • This means that the standard deviation is also a measure of the spread, only that it increases a little more slowly than the variance. If you know the variance, it can easily be converted into the standard deviation (and vice versa).

\ begin {align *}
\ sigma = \ sqrt {V (X)} = \ sqrt {\ sigma ^ 2} \ notag
\ end {align *}

Basic tasks

  • Calculate and interpret expected value and standard deviation
  • Calculate the probability that the random variable will take on values ​​that deviate from the expected value by specified values.

example

A teacher wants to know how his students are doing and how much the good or bad students deviate from the average. The following table shows the distribution of grades

$
\ begin {array} {c | c | c | c | c | c | c}
\ text {Note} \ x_j & 1 & 2 & 3 & 4 & 5 & 6 \ \ hline
p_j & 0.1 & 0.15 & 0.5 & 0.2 & 0 & 0.05
\ end {array}
$

The expected value is calculated as follows:

\ begin {align *}
\ mu = E (x) = 1 \ times 0.1 + 2 \ times 0.15 + 3 \ times 0.5 + 4 \ times 0.2 + 5 \ times 0 + 6 \ times 0.05 = 3 \ notag
\ end {align *}

Interpretation: The grade point average is $ 3 $.

We calculate the standard deviation using the variance:

\ begin {align *}
\ sigma ^ 2 & = (1-3) ^ 2 \ times 0.1 + (2-3) ^ 2 \ times 0.15 + (3-3) ^ 2 \ times 0.5 + (4-3) ^ 2 \ times 0.2 \ notag \ & + (5-3) ^ 2 \ times 0 + (6-3) ^ 2 \ times 0.05 = 1.2 \ notag \
\ Rightarrow \ quad \ sigma & = \ sqrt {1,2} \ notag
\ end {align *}

The spread around the grade point average is low. A stick diagram would make this further clear. With such tasks you often have to compare several scenarios and say which scenario spreads more.

Our Mathe-Abi'21 learning booklets Explanations ✔ Examples ✔ Free learning videos ✔

New!