Relationship between two random variables having same expectation

In summary, when two random variables have the same expectation, it means they have an equal chance of taking on values around their average value. This can help in understanding their relationship and making predictions. The correlation between these variables can range from -1 to 1, indicating their linear relationship. Two random variables with the same expectation can have different distributions, which can affect their behavior together. The covariance between these variables can measure how they vary together and their strength and direction of relationship. The relationship between two random variables with the same expectation can also change over time due to various factors, highlighting the importance of continuous analysis.
  • #1
omaradib
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Homework Statement



Say, it is known that
[tex]E_X[f(X)] = E_X[g(X)] = a[/tex] where [tex]f(X)[/tex] and [tex]g(X)[/tex] are two functions of the same random variable [tex]X[/tex]. What is the relationship between [tex]f(X)[/tex] and [tex]g(X)[/tex]?

Homework Equations





The Attempt at a Solution



My answer is [tex]f(X) = g(X) + h(X)[/tex] where [tex]E_X[h(X)] = 0[/tex].

This relationship is apparent by taking expectation in both sides. But, is it necessarily and sufficiently true?
 
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  • #2
Anyone please?
 

1. What is the significance of two random variables having the same expectation?

The expectation of a random variable is a measure of its average value. When two random variables have the same expectation, it means that they have an equal chance of taking on values around this average. This can be useful in understanding the relationship between the two variables and making predictions based on their joint behavior.

2. How does the correlation between two random variables with the same expectation affect their relationship?

The correlation between two random variables measures their linear relationship, or how closely their values are related to each other. When two random variables have the same expectation, their correlation can range from -1 to 1. A correlation of 0 means there is no linear relationship between the variables, while a correlation of 1 or -1 means there is a perfect positive or negative linear relationship.

3. Can two random variables with the same expectation have different distributions?

Yes, two random variables can have the same expectation but different distributions. This means that although they have an equal chance of taking on values around the average, the spread or shape of their distributions may be different. This can impact their relationship and how they behave together.

4. How can the covariance between two random variables with the same expectation be interpreted?

The covariance between two random variables measures how they vary together. When the covariance is positive, it means that the variables tend to have similar values at the same time. When the covariance is negative, it means that the variables tend to have opposite values. When two random variables have the same expectation, their covariance can help us understand the strength and direction of their relationship.

5. Can the relationship between two random variables with the same expectation change over time?

Yes, the relationship between two random variables can change over time due to various factors such as external influences or changes in the underlying processes. Even though the variables may have the same expectation, their joint behavior and correlation can vary over time. This highlights the importance of continuously analyzing and understanding the relationship between two random variables.

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