Continuous Random Variables - Definition | Brilliant Math & Science ...

Recall that a random variable is a quantity which is drawn from a statistical distribution, i.e. it does not have a fixed value. A continuous random variable is a random variable whose statistical distribution is continuous. Formally:

A continuous random variable is a function \(X\) on the outcomes of some probabilistic experiment which takes values in a continuous set \(V\).

That is, the possible outcomes lie in a set which is formally (by real-analysis) continuous, which can be understood in the intuitive sense of having no gaps. The fact that \(X\) is technically a function can usually be ignored for practical purposes outside of the formal field of measure theory. In applications, \(X\) is treated as some quantity which can fluctuate e.g. in repeated experiments, which has statistical properties like mean and variance .

In the next article on continuous probability density functions, the meaning of \(X\) will be explored in a more practical setting.

Which of the following are continuous random variables?

(1) The sum of numbers on a pair of two dice.

(2) The possible sets of outcomes from flipping ten coins.

(3) The possible sets of outcomes from flipping (countably) infinite coins.

(4) The possible values of the temperature outside on any given day.

(5) The possible times that a person arrives at a restaurant.

Solution:

(4) and (5) are the continuous random variables. Going through each case in order:

(1) Ignoring reordering of the dice and repeated values, there are a maximum of 36 possible sets of values on the two dice. In reality, the number is less than this, but would require more careful counting. However, this is sufficent to note that this value is a discrete random variable, since the number of possible values is finite.

(2) Again, the possible sets of outcomes is larger (bounded above by \(2^{10}\), certainly) but finite and the same logic applies as in (1).

(3) This case is more interesting because there are infinitely many coins. However, there are only countably many sets of outcomes. A countable set of real numbers is not continuous (consider the countable rational numbers, which are not continuous).

(4) The temperature outside on any given day could be any real number in a given reasonable range. In particular, on no two days is the temperature exactly the same number out to infinite decimal places. Thus, the temperature takes values in a continuous set.

(5) This case is similar to (4): no two people ever arrive at exactly the same time out to infinite precision. The precise time a person arrives is a value in the set of real numbers, which is continuous. Note that this implies that the probability of arriving at any one given time is zero, a fact which will be discussed in the next article.

The minimum outcome from rolling infinitely many dice The number of raindrops in a storm The number of people that show up to class The angle you face after spinning in a circle Reveal the answer

Which of the following answers is the continuous random variable?

The correct answer is: The angle you face after spinning in a circle