Proving The Law Of The Unconscious Statistician

Introduction

The Law of the Unconscious Statistician is a fundamental concept in probability theory that allows us to compute the expected value of a function of a random variable. In this article, we will prove this law, which states that the expected value of a function of a random variable is equal to the sum of the function evaluated at each possible value of the random variable, weighted by the probability of each value. We will start by defining the problem and the necessary notation, and then proceed to the proof.

Notation and Definitions

Let $X$ be a discrete random variable with probability mass function $f_x(x)$, and let $g(x)$ be a function of $X$. We want to prove that the expected value of $g(X)$ is equal to the sum of $g(x)$ evaluated at each possible value of $X$, weighted by the probability of each value. Mathematically, we want to prove that:

$$\mathbb E(Y) = \sum_x g(x)f_x(x)$$

where $Y = g(X)$.

Proof

To prove this law, we can start by writing the definition of the expected value of $Y$:

$$\mathbb E(Y) = \sum_y yf_y(y)$$

where $f_y(y)$ is the probability mass function of $Y$. Since $Y = g(X)$, we can substitute $y = g(x)$ into the above equation:

$$\mathbb E(Y) = \sum_x g(x)f_y(g(x))$$

Now, we can use the fact that the probability mass function of $Y$ is equal to the probability mass function of $X$ evaluated at the image of $X$ under $g$:

$$f_y(g(x)) = f_x(x)$$

Substituting this into the above equation, we get:

$$\mathbb E(Y) = \sum_x g(x)f_x(x)$$

This is the desired result.

Discussion

The proof of the Law of the Unconscious Statistician is quite straightforward, and it relies on the definition of the expected value and the properties of the probability mass function. However, the law itself is a powerful tool in probability theory, and it has many applications in statistics and engineering.

Example

To illustrate the Law of the Unconscious Statistician, let's consider an example. Suppose we have a random variable $X$ that takes on the values $0$ and $1$ with equal probability, and let $g(x) = x^2$. We want to compute the expected value of $g(X)$.

Using the Law of the Unconscious Statistician, we can write:

$$\mathbb E(g(X)) = \sum_x g(x)f_x(x) = g(0)f_x(0) + g(1)f_x(1)$$

Since $f_x(0) = f_x(1) = 1/2$, we get:

$$\mathbb E(g(X)) = (0^2)(1/2) + (1^2)(1/2) = 1/2$$

This is the expected value of $g(X)$.

Conclusion

In this article, we have proved the Law of the Unconscious Statistician, which states that the expected value of a function of a random variable is equal to the sum of the function evaluated at each possible value of the random variable, weighted by the probability of each value. We have also illustrated the law with an example, and we have discussed its importance in probability theory.

References

• [1] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and random processes. Oxford University Press.
• [2] Ross, S. M. (2014). Introduction to probability models. Academic Press.

Further Reading

For further reading on the Law of the Unconscious Statistician, we recommend the following resources:

• [1] Wikipedia: Law of the unconscious statistician
• [2] Khan Academy: Expected value and variance
• [3] MIT OpenCourseWare: Probability and statistics

Glossary

• Expected value: The average value of a random variable.
• Probability mass function: A function that assigns a probability to each possible value of a discrete random variable.
• Law of the unconscious statistician: A fundamental concept in probability theory that allows us to compute the expected value of a function of a random variable.
  Q&A: The Law of the Unconscious Statistician =====================================================

Introduction

In our previous article, we proved the Law of the Unconscious Statistician, which states that the expected value of a function of a random variable is equal to the sum of the function evaluated at each possible value of the random variable, weighted by the probability of each value. In this article, we will answer some common questions about the Law of the Unconscious Statistician and provide additional insights into its application.

Q: What is the Law of the Unconscious Statistician?

A: The Law of the Unconscious Statistician is a fundamental concept in probability theory that allows us to compute the expected value of a function of a random variable. It states that the expected value of a function of a random variable is equal to the sum of the function evaluated at each possible value of the random variable, weighted by the probability of each value.

Q: When can I use the Law of the Unconscious Statistician?

A: You can use the Law of the Unconscious Statistician whenever you need to compute the expected value of a function of a random variable. This is a common scenario in statistics and engineering, where you may need to compute the expected value of a function of a random variable to make decisions or predictions.

Q: What are the assumptions of the Law of the Unconscious Statistician?

A: The Law of the Unconscious Statistician assumes that the random variable is discrete and that the function is well-defined. It also assumes that the probability mass function of the random variable is known.

Q: How do I apply the Law of the Unconscious Statistician?

A: To apply the Law of the Unconscious Statistician, you need to follow these steps:

1. Define the function of the random variable that you want to compute the expected value of.
2. Identify the possible values of the random variable and their corresponding probabilities.
3. Evaluate the function at each possible value of the random variable.
4. Weight each value of the function by the probability of the corresponding value of the random variable.
5. Sum up the weighted values to get the expected value of the function.

Q: What are some common applications of the Law of the Unconscious Statistician?

A: The Law of the Unconscious Statistician has many applications in statistics and engineering, including:

• Computing the expected value of a function of a random variable to make decisions or predictions.
• Modeling the behavior of complex systems, such as financial markets or supply chains.
• Analyzing the performance of algorithms or systems, such as machine learning models or computer networks.

Q: What are some common mistakes to avoid when using the Law of the Unconscious Statistician?

A: Some common mistakes to avoid when using the Law of the Unconscious Statistician include:

• Failing to check the assumptions of the law, such as the discreteness of the random variable and the well-definedness of the function.
• Using the law with a continuous random variable, which is not allowed.
• Failing to evaluate the function at each possible value of the random variable.
• Failing to weight each value of the function by the probability of the corresponding value of the random variable.

Conclusion

In this article, we have answered some common questions about the Law of the Unconscious Statistician and provided additional insights into its application. We hope that this article has been helpful in clarifying the law and its assumptions, and that it has provided you with a better understanding of how to apply it in practice.

References

• [1] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and random processes. Oxford University Press.
• [2] Ross, S. M. (2014). Introduction to probability models. Academic Press.

Further Reading

For further reading on the Law of the Unconscious Statistician, we recommend the following resources:

• [1] Wikipedia: Law of the unconscious statistician
• [2] Khan Academy: Expected value and variance
• [3] MIT OpenCourseWare: Probability and statistics

Glossary

• Expected value: The average value of a random variable.
• Probability mass function: A function that assigns a probability to each possible value of a discrete random variable.
• Law of the unconscious statistician: A fundamental concept in probability theory that allows us to compute the expected value of a function of a random variable.