Question

Given below are two statements:
Statement I: A random variable that assume a infinite or a unaccountably infinite number of values is called continuous random variable.
Statement II: A random variable is called discrete if it has either a finite or a countable number of possible values.
In light of the above statements, choose the most appropriate answer:

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but Statement II is incorrect
• D. Statement I is incorrect but Statement II is correct

Hint:

Remember: "Countable" (finite or infinite) implies a discrete variable (e.g., number of rainy days).
"Uncountable" implies a continuous variable (e.g., actual depth of daily rainfall).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A random variable is a mathematical rule that assigns a real numerical value to each outcome in a sample space.
Based on the nature of the values it can assume, a random variable is classified as either discrete or continuous.

Step 2: Detailed Explanation:
Let us analyze both statements:
- Statement I: A continuous random variable is defined as one that can assume any real value within a specified interval or set of intervals.
Since the set of real numbers within any interval is uncountably infinite, the random variable can assume an unaccountably infinite number of possible values.
Thus, Statement I is mathematically correct.
- Statement II: A discrete random variable is one that can take on only a finite number of distinct values, or a countably infinite sequence of values (such as the set of natural numbers).
Because its possible values can be listed or put in a one-to-one correspondence with the set of integers, they are countable.
Thus, Statement II is also mathematically correct.
Since both statements are correct, we select the corresponding option.

Step 3: Final Answer:
Both Statement I and Statement II are correct.