# Random variables

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Random variables

A random variable is variable that gets its value by some random, (stochastic) process. A random variable is usually denoted by a capital letter, and usually in italics.

We may for example have

• X=the result after throwing a fair six sided die.
• Y=the number of red cars passing a certain crossing in one day.
• Z=the sum of the values of the card in a poker hand, (A=1,… K=12)
• W=2 + the result after throwing a fair six sided die.

The result of an experiment (that could be an actual experiment, or, say,  a survey)  is called an outcome of the experiment. If I throw a die, and get a six, then six is the outcome of the experiment.

The set of possible outcomes is called the sample space sample space of the random variable.  For a ordinary fair dice the sample space is:

{1 ,2, 3, 4, 5, 6}

Capital X vs. non capital x

• X=a random variable.

This is basically connected to a description of how to get the random value, as described above. We may for example use this to ask “what is the probability that X=6?”, i.e. what is the probability that outcome of throwing a die is 6? This is often written P(X=6)

• x=an actual outcome.

We may for example ask what is P(X=x) , i.e, what is the probability that our random variable X has the value x. This is often used in formulas for the probability. If we for example would have   an urn with marbles marked with numbers, four with a 1, three with a 2, two with a 3 and one with a 4, and X is the result of randomly selecting one marble (after replacement)  then we have the formula

$\text{P}\left( X=x \right)=\left\{ \begin{matrix} \frac{5-x}{10},\quad \text{if}\ 1\le x\le 4 \\ 0,\quad \quad \text{else}\text{.}\quad \\ \end{matrix} \right.$

I.e. the probability to get say a 3 is (5-3)/10=2/10, just as expected.

Up a level : Probability and Statistics
Next page : The Mean or AverageLast modified: Nov 11, 2017 @ 18:38