Expectation of Ratio of Two Identically Distributed Random Variables is Greater Than 1

Oct 13, 2025 by leolavanderia.com.br 35 views

Expectation of Ratio of Two Identically Distributed Random Variables is Greater Than 1

In probability theory, the expectation of a random variable is a fundamental concept that plays a crucial role in understanding the behavior of random phenomena. When dealing with ratios of random variables, it is essential to consider the properties of the expectation. In this article, we will explore the expectation of the ratio of two identically distributed random variables and show that it is greater than 1.

The Cauchy-Schwarz inequality is a powerful tool in probability theory that provides a lower bound for the expectation of the product of two random variables. For two random variables X and Y, the Cauchy-Schwarz inequality states that:

(E[XY])^2 \leq E[X^2].E[Y^2]

This inequality can be used to derive a lower bound for the expectation of the ratio of two random variables.

Let X and Y be two identically distributed (i.d.) positive random variables. If they are furthermore independent, we can use the Cauchy-Schwarz inequality to derive a lower bound for the expectation of the ratio X/Y.

From the Cauchy-Schwarz inequality, we have:

(E[XY])^2 \leq E[X^2].E[Y^2]

Since X and Y are identically distributed, we have:

E[X^2] = E[Y^2]

Substituting this into the Cauchy-Schwarz inequality, we get:

(E[XY])^2 \leq E[X^2]^2

Taking the square root of both sides, we get:

E[XY] \leq E[X^2]

Now, let's consider the expectation of the ratio X/Y. We can write:

E[X/Y] = E[X].E[1/Y]

Since X and Y are independent, we have:

E[X].E[1/Y] = E[X]

Substituting this into the previous equation, we get:

E[X/Y] = E[X]

To show that the expectation of the ratio X/Y is greater than 1, we need to find a lower bound for E[X/Y]. From the previous equation, we have:

E[X/Y] = E[X]

Since X is a positive random variable, we have:

E[X] \geq 1

Therefore, we can conclude that:

E[X/Y] \geq 1

This shows that the expectation of the ratio X/Y is greater than 1.

In this article, we have shown that the expectation of the ratio of two identically distributed random variables is greater than 1. This result is a consequence of the Cauchy-Schwarz inequality and the properties of the expectation. The expectation of the ratio X/Y is a fundamental concept in probability theory, and this result has important implications for understanding the behavior of random phenomena.

The result that the expectation of the ratio X/Y is greater than 1 has important implications for various fields, including finance, economics, and engineering. For example, in finance, this result can be used to derive lower bounds for the expected return on investment. In economics, this result can be used to understand the behavior of economic systems and make predictions about future outcomes. In engineering, this result can be used to design and optimize systems that involve random variables.

There are several directions for future research on this topic. One area of research is to explore the properties of the expectation of the ratio of two random variables when the random variables are not identically distributed. Another area of research is to investigate the implications of this result for various fields, including finance, economics, and engineering.

[1] Cauchy-Schwarz inequality
[2] Expectation of a random variable
[3] Identically distributed random variables

The appendix contains the proof of the Cauchy-Schwarz inequality.

The Cauchy-Schwarz inequality can be proved using the following steps:

Let X and Y be two random variables.
Define the function f(x) = xY(x).
Use the properties of the expectation to derive a lower bound for E[f(x)].
Use the lower bound for E[f(x)] to derive the Cauchy-Schwarz inequality.

The details of the proof are as follows:

Let X and Y be two random variables.
Define the function f(x) = xY(x).
We have:

E[f(x)] = E[xY(x)]

Using the properties of the expectation, we can write:

E[f(x)] = \int_{-\infty}^{\infty} xY(x) dF(x)

where F(x) is the cumulative distribution function of X.

Using the Cauchy-Schwarz inequality for integrals, we have:

\left( \int_{-\infty}^{\infty} xY(x) dF(x) \right)^2 \leq \int_{-\infty}^{\infty} x^2 dF(x) \int_{-\infty}^{\infty} Y^2(x) dF(x)

Simplifying the right-hand side, we get:

\left( \int_{-\infty}^{\infty} xY(x) dF(x) \right)^2 \leq E[X^2].E[Y^2]

This is the Cauchy-Schwarz inequality.

The Cauchy-Schwarz inequality can be generalized to the case where X and Y are random vectors. In this case, the inequality becomes:

(E[XY])^2 \leq E[X^T X].E[Y^T Y]

where X and Y are random vectors, and X^T and Y^T are their transpose.
Q&A: Expectation of Ratio of Two Identically Distributed Random Variables

In our previous article, we explored the expectation of the ratio of two identically distributed random variables and showed that it is greater than 1. In this article, we will answer some frequently asked questions about this topic.

A: Identically distributed random variables are random variables that have the same probability distribution. In other words, if X and Y are identically distributed, then their probability density functions (PDFs) are the same.

A: The Cauchy-Schwarz inequality is a mathematical inequality that provides a lower bound for the expectation of the product of two random variables. It states that:

(E[XY])^2 \leq E[X^2].E[Y^2]

A: The expectation of the ratio of two identically distributed random variables is related to the Cauchy-Schwarz inequality through the following equation:

E[X/Y] = E[X].E[1/Y]

Using the Cauchy-Schwarz inequality, we can show that:

E[X/Y] \geq 1

A: This result has important implications for various fields, including finance, economics, and engineering. For example, in finance, this result can be used to derive lower bounds for the expected return on investment. In economics, this result can be used to understand the behavior of economic systems and make predictions about future outcomes. In engineering, this result can be used to design and optimize systems that involve random variables.

A: No, this result cannot be generalized to the case where the random variables are not identically distributed. The Cauchy-Schwarz inequality only holds for identically distributed random variables.

A: Some common applications of this result include:

Deriving lower bounds for the expected return on investment in finance
Understanding the behavior of economic systems and making predictions about future outcomes in economics
Designing and optimizing systems that involve random variables in engineering

A: Some common misconceptions about this result include:

Thinking that the expectation of the ratio of two random variables is always greater than 1
Thinking that the Cauchy-Schwarz inequality only holds for random variables with a specific distribution
Thinking that this result has no practical applications

In this article, we have answered some frequently asked questions about the expectation of the ratio of two identically distributed random variables. We have shown that this result has important implications for various fields and has many practical applications.

Q: What are identically distributed random variables?
A: Identically distributed random variables are random variables that have the same probability distribution.
Q: What is the Cauchy-Schwarz inequality?
A: The Cauchy-Schwarz inequality is a mathematical inequality that provides a lower bound for the expectation of the product of two random variables.
Q: How is the expectation of the ratio of two identically distributed random variables related to the Cauchy-Schwarz inequality?
A: The expectation of the ratio of two identically distributed random variables is related to the Cauchy-Schwarz inequality through the following equation: E[X/Y] = E[X].E[1/Y].
Q: What are the implications of this result?
A: This result has important implications for various fields, including finance, economics, and engineering.
Q: Can this result be generalized to the case where the random variables are not identically distributed?
A: No, this result cannot be generalized to the case where the random variables are not identically distributed.
Q: What are some common applications of this result?
A: Some common applications of this result include deriving lower bounds for the expected return on investment in finance, understanding the behavior of economic systems and making predictions about future outcomes in economics, and designing and optimizing systems that involve random variables in engineering.
Q: What are some common misconceptions about this result?
A: Some common misconceptions about this result include thinking that the expectation of the ratio of two random variables is always greater than 1, thinking that the Cauchy-Schwarz inequality only holds for random variables with a specific distribution, and thinking that this result has no practical applications.

[1] Cauchy-Schwarz inequality
[2] Expectation of a random variable
[3] Identically distributed random variables