Which of the following sample spaces would satisfy the definition of a continuous random variable?

• A. The set of all real numbers between 0 and 1, that is, $\{x:0<x<1\}$
• B. The set of all integers from 1 to 10, that is, $\{1,2,\dots ,10\}$
• C. The set of days in a week: $\{\mathrm{Monday},\mathrm{Tuesday},\dots ,\mathrm{Sunday}\}$
• D. The set of outcomes when flipping a coin: $\{H,T\}$

Answer: (A)

Verified step by step guidance

1. Recall the definition of a continuous random variable: it takes values from an uncountably infinite set, typically intervals of real numbers, where the variable can assume any value within that range.
2. Examine each sample space to determine if it is continuous or discrete. A continuous sample space contains all real numbers within an interval, while a discrete sample space contains countable or finite distinct outcomes.
3. The set of all real numbers between 0 and 1, written as \(\{x : 0 < x < 1\}\), is an interval containing infinitely many values with no gaps, which fits the definition of a continuous sample space.
4. The set of all integers from 1 to 10, \(\{1, 2, \ldots, 10\}\), is a finite set of distinct points, so it is discrete, not continuous.
5. The set of days in a week and the outcomes of flipping a coin are also finite discrete sets, so they do not satisfy the definition of a continuous random variable.