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Random variable

variable whose possible values are numerical outcomes of a random phenomenon

A random variable is used in mathematics to study probability theory. It was developed to check out the chance of events happening or in games.

Contents

DefinitionEdit

Take two measurable spaces, and name them   and  . A measurable space is any pair of sets, called   and  , that follow these rules:

  1.   is not empty;
  2. The elements of   are subsets of  ;
  3.   and the empty set are both elements of  ;
  4.   is closed under complements and countable unions.

A random variable, named  , is a measurable function from   to  . This is written  .

The set   is called the sample space. The set   is called the event space.

Use in ProbabilityEdit

Take a probability measure   on  . Take a set   in  . Then   is defined to mean  .

Examples with DiceEdit

When you roll a dice, 6 events can happen. These events are the different numbers that can show on the dice: 1, 2, 3, 4, 5, or 6.

Dicing FruitEdit

Here is an example of a random variable. You roll a fair dice once. If the number on the dice is odd, you eat an apple. If the number is even, you eat an orange. The random variable is the type of fruit that you will eat. Before you roll the dice, you don't know if you will eat an apple or an orange. We can also write this with mathematics:

Take  . Take   to be the power set of  . Take  . Take   to be the power set of  . Our random variable follows the even/odd rule from above:

 

Here   represents the number on the dice after you roll it. We said the dice is fair. In mathematics, this means   for   and  .

The event that you eat an apple is a set in the event space. It is  . The probability that you eat an apple is the probability measure of this set. It is  .

Dicing Fruit UnfairlyEdit

Here is another example. Sometimes a dice is unbalanced. This means that it rolls some numbers more often than others. We can do the same experiment as above, but with an unbalanced dice.

For example, now use a dice that has been changed in one way: it never rolls the number 5. Then you are less likely to eat an apple, because 5 is an odd number. The random variable is exactly the same as before:

 

This is because end possibilities haven't changed. After the dice is rolled, you will eat an apple or an orange. You don't know which. What has changed is the probability that you eat an apple or an orange. In mathematics, this means

 

The event that you eat an apple is the same set as for the fair dice. It is  . But this event is now less likely to happen, because the dice cannot roll a 5. In other words, the probability that you eat an apple is different than for the fair dice. It is  .

Probability SpacesEdit

The examples show that a random variable doesn't automatically give probabilities. If we choose   differently, one random variable can give different probabilities. This can be confusing. For this reason, mathematicians often define a random variable on a probability space. The mathematical notation for this is  . This fixes the probability measure. Then there is no confusion about the probabilities of events.

Further readingEdit

  • Klenke, Achim, Probability Theory: A Comprehensive Course : Springer (2013) ISBN 978-1-4471-5360-3