Average of function vs. function evaluated at average

In summary, The conversation discusses a function f(x) of a random variable x with average m and probability distribution p(x). The question is whether the average of f(x) is greater than f(m), and if there are any theorems that can help determine this. It is mentioned that the concaveness of the function and shape of the distribution may play a role, but it is also noted that if the function is concave, it does not depend on the shape of p. Jensen's inequality is mentioned as a potential tool for answering the question.
  • #1
nigeisel
1
0
Dear everyone,
consider a function f(x) of a random variable x with average m and probability distribution p(x).

I would like to know under which conditions the average of f(x) is greater than f(m), i.e., under which conditions is the average of the function greater than the function evaluated at the average value of the random variable.

\int p(x) f(x) dx > f(m) ?

Does anyone know about theorems that might help me? I suppose it depends on the concaveness of the function f and shape of the distribution p.

I would be very grateful for help.
Nico
 
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  • #2
If the function is concave, it does not depend on tha shape of p. See http://en.wikipedia.org/wiki/Jensen%27s_inequality"
 
Last edited by a moderator:
  • #3
In fact, that property is often used as the very definition of concave, convex, and linear.
 

1. What is the difference between average of function and function evaluated at average?

The average of a function is the average value of the function over a given interval, while the function evaluated at average is the value of the function at the average of the interval. In other words, the average of a function takes into account all points within the interval, while the function evaluated at average only considers the midpoint of the interval.

2. Which method is more accurate in calculating the average value of a function?

The average of a function is generally more accurate in calculating the average value, as it considers a larger number of points within the interval. However, if the function is relatively smooth and has a consistent slope, the function evaluated at average method may also yield accurate results.

3. Can the average of a function and the function evaluated at average yield different results?

Yes, the average of a function and the function evaluated at average can yield different results. This is because the average of a function takes into account all points within the interval, while the function evaluated at average only considers the midpoint of the interval. Therefore, if the function has a lot of variation within the interval, the two methods may yield different results.

4. In which situations would it be more appropriate to use the average of a function method?

The average of a function method is more appropriate in situations where the function has a lot of variation within the interval. This method takes into account all points within the interval, providing a more accurate representation of the average value of the function.

5. How can the average of a function and the function evaluated at average be used in real-world applications?

The average of a function and the function evaluated at average are commonly used in fields such as economics, physics, and engineering to calculate and analyze data. For example, the average of a function can be used to determine the average temperature over a period of time, while the function evaluated at average can be used to calculate the average velocity of an object over a given distance.

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