A continuous random variable is one that can take an infinite number of values within a given range. It is typically represented by an interval on the real number line.

Let's analyze each sample space to determine if it satisfies the definition of a continuous random variable:

1. $X = \{\text{the number of days in the week that have the letter "r"}\}$:

   • This sample space consists of discrete values (specific integers corresponding to the number of days).
   • Example values might be {0, 1, 2, 3, 4, 5, 6, 7}.
   • Conclusion: Not continuous.

2. $X = \{\text{odd integers between 5 and 15}\}$:

   • This sample space consists of discrete values (specific odd integers).
   • Example values are {7, 9, 11, 13, 15}.
   • Conclusion: Not continuous.

3. $X = [0,500]$:

   • This sample space represents all real numbers between 0 and 500, inclusive.
   • It includes an infinite number of possible values within the interval.
   • Conclusion: Continuous.

4. $X = \{0,1,2,3\}$:

   • This sample space consists of discrete values (specific integers).
   • Example values are {0, 1, 2, 3}.
   • Conclusion: Not continuous.

Therefore, the sample space $X = [0,500]$ satisfies the definition of a continuous random variable.

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:

1. What are the key differences between discrete and continuous random variables?
2. Can a continuous random variable take on any value within an interval, including irrational numbers?
3. How is the probability density function (PDF) related to continuous random variables?
4. What are some common examples of continuous random variables in real life?
5. How do you calculate probabilities for continuous random variables?
6. Can a random variable be both discrete and continuous?
7. How are cumulative distribution functions (CDFs) used with continuous random variables?
8. What is the importance of the area under the curve in the context of continuous random variables?

Tip: When dealing with continuous random variables, always consider the entire range of possible values within the interval to accurately understand and analyze the variable's behavior.