Which Of These Is An Example Of A Discrete Random Variable?A. Time To Complete A Triathlon B. Height Of A Tree C. Weight Of A Hippopotamus D. First Digit Of A Zip Code

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In probability theory, a random variable is a function that assigns a numerical value to each outcome of a random experiment. There are two main types of random variables: discrete and continuous. In this article, we will explore which of the given options is an example of a discrete random variable.

What is a Discrete Random Variable?

A discrete random variable is a random variable that can take on a countable number of distinct values. In other words, it is a variable that can only take on a specific set of values, and these values are usually whole numbers. Examples of discrete random variables include the number of heads in a coin toss, the number of students in a class, and the number of defects in a manufacturing process.

Examples of Discrete Random Variables

Let's consider the options given in the question:

A. Time to complete a triathlon: This is an example of a continuous random variable. The time to complete a triathlon can take on any value within a certain range, and it is not limited to a specific set of values.

B. Height of a tree: This is also an example of a continuous random variable. The height of a tree can take on any value within a certain range, and it is not limited to a specific set of values.

C. Weight of a hippopotamus: This is an example of a continuous random variable. The weight of a hippopotamus can take on any value within a certain range, and it is not limited to a specific set of values.

D. First digit of a zip code: This is an example of a discrete random variable. The first digit of a zip code can only take on a specific set of values, which are the digits 0-9. This is a countable number of distinct values, making it a discrete random variable.

Why is the First Digit of a Zip Code a Discrete Random Variable?

The first digit of a zip code is a discrete random variable because it can only take on a specific set of values. In the United States, the first digit of a zip code is usually between 0 and 9, and it is not possible for the first digit to be any other value. This is a countable number of distinct values, making it a discrete random variable.

Conclusion

In conclusion, the first digit of a zip code is an example of a discrete random variable. It is a countable number of distinct values, and it is not possible for the first digit to be any other value. This is in contrast to the other options, which are examples of continuous random variables.

Key Takeaways

  • A discrete random variable is a random variable that can take on a countable number of distinct values.
  • Examples of discrete random variables include the number of heads in a coin toss, the number of students in a class, and the number of defects in a manufacturing process.
  • The first digit of a zip code is an example of a discrete random variable because it can only take on a specific set of values.
  • Continuous random variables, on the other hand, can take on any value within a certain range.

Further Reading

If you are interested in learning more about discrete and continuous random variables, I recommend checking out the following resources:

References

In our previous article, we explored the concept of discrete random variables and provided examples of how they can be used to model real-world phenomena. In this article, we will answer some frequently asked questions about discrete random variables.

Q: What is the difference between a discrete random variable and a continuous random variable?

A: A discrete random variable is a random variable that can take on a countable number of distinct values, whereas a continuous random variable can take on any value within a certain range.

Q: Can a discrete random variable take on any value within a certain range?

A: No, a discrete random variable can only take on a specific set of values. For example, the number of heads in a coin toss can only take on the values 0, 1, or 2.

Q: How do I determine if a random variable is discrete or continuous?

A: To determine if a random variable is discrete or continuous, you need to consider the possible values it can take on. If the variable can only take on a specific set of values, it is a discrete random variable. If the variable can take on any value within a certain range, it is a continuous random variable.

Q: Can a discrete random variable be used to model a continuous phenomenon?

A: Yes, a discrete random variable can be used to model a continuous phenomenon by approximating the continuous variable with a discrete variable. For example, the height of a person can be approximated by a discrete variable that takes on the values 5, 5.5, 6, etc.

Q: What are some common examples of discrete random variables?

A: Some common examples of discrete random variables include:

  • The number of heads in a coin toss
  • The number of students in a class
  • The number of defects in a manufacturing process
  • The first digit of a zip code
  • The number of successes in a binomial experiment

Q: Can a discrete random variable be used to model a real-world phenomenon that has a large number of possible values?

A: Yes, a discrete random variable can be used to model a real-world phenomenon that has a large number of possible values. For example, the number of possible outcomes in a game of chance can be modeled using a discrete random variable.

Q: How do I calculate the probability of a discrete random variable?

A: To calculate the probability of a discrete random variable, you need to use the formula:

P(X = x) = (Number of outcomes where X = x) / (Total number of possible outcomes)

For example, if you roll a fair six-sided die, the probability of rolling a 4 is:

P(X = 4) = (Number of outcomes where X = 4) / (Total number of possible outcomes) = 1 / 6

Q: Can a discrete random variable be used to model a real-world phenomenon that has a small number of possible values?

A: Yes, a discrete random variable can be used to model a real-world phenomenon that has a small number of possible values. For example, the number of possible outcomes in a game of chance with a small number of possible outcomes can be modeled using a discrete random variable.

Q: How do I use a discrete random variable to model a real-world phenomenon?

A: To use a discrete random variable to model a real-world phenomenon, you need to:

  1. Identify the possible values of the variable
  2. Assign a probability to each value
  3. Use the variable to model the phenomenon

For example, if you want to model the number of students in a class, you can use a discrete random variable that takes on the values 10, 20, 30, etc. You can then assign a probability to each value based on the actual number of students in the class.

Conclusion

In conclusion, discrete random variables are a powerful tool for modeling real-world phenomena. By understanding the properties and applications of discrete random variables, you can use them to model a wide range of phenomena, from the number of heads in a coin toss to the number of students in a class.

Key Takeaways

  • A discrete random variable is a random variable that can take on a countable number of distinct values.
  • Examples of discrete random variables include the number of heads in a coin toss, the number of students in a class, and the number of defects in a manufacturing process.
  • A discrete random variable can be used to model a real-world phenomenon that has a large number of possible values.
  • The probability of a discrete random variable can be calculated using the formula P(X = x) = (Number of outcomes where X = x) / (Total number of possible outcomes).

Further Reading

If you are interested in learning more about discrete random variables, I recommend checking out the following resources:

References