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Continuous random variables: basic ideas

Random variables have been introduced in the module Discrete probability distributions . Recall that a random variable is a variable whose value is determined by the outcome of a random procedure.

There are two main types of random variables: discrete and continuous. The modules Discrete probability distributions and Binomial distribution deal with discrete random variables; we now turn our attention to the second type, continuous random variables.

A continuous random variable is one that can take any real value within a specified range.

A discrete random variable takes some values and not others; we cannot obtain a value of 4.73 when rolling a fair die. By contrast, a continuous random variable can take any value, in principle, within a specified range.

We have already seen examples of continuous random variables, when the idea of a random variable was first introduced.

Example: Five people born in 1995

Five babies born in 1995 are followed up over their lives, and major health and milestone events are recorded. Here are two continuous random variables that could be defined:

Let \(W\) be the average height of the five people at age 18. Then the value of \(W\) must be positive, but there is no obvious upper bound. The common practice in such cases is to say that the possible values are \(W > 0\); we will assign extremely low probabilities to large values.
Let \(T_i\) be the total time spent on Facebook by individual \(i\) up to age 18. Then \(T_i\) in this case is limited by the total time span being considered. If we measure \(T_i\) in years, then \(0 \leq T_i \leq 18\); again, values anywhere near the logical maximum of 18 years will be assigned essentially zero probability.

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