# Is an expected value of fx(x) the same as this expected value?

• fateswarm
In summary, an expected value is a predicted or average value of a random variable, calculated by multiplying each possible outcome by its probability and summing the results. To calculate an expected value, you multiply each possible outcome by its probability, then add all the results together. This is often represented as E(x) = ∑xP(x), where x is the possible outcome and P(x) is its probability. The expected value and the mean are essentially the same concept, representing the average outcome of a random variable. However, the expected value can also be calculated for non-numerical variables, while the mean only applies to numerical data. An expected value can be negative if the possible outcomes of a random variable have negative values and their probabilities

#### fateswarm

If a probability density function is fx(x) = 5x^4 for 0<x<1 and 0 elsewhere, then the expected value of E[x^0.5] is the integral of ∫x^0.5*fx(x) between 0 and 1 which is 10/11.

However if I try to approach it as the expected value of Y=X^0.5 then I find fy(y) = 5*lny/y^4 and its expected value is 20/33.

What's wrong? Is it a right approach or are those two things different?

Ah, I got it, x is y^2.. not the other way around.

## 1. What is an expected value?

An expected value is a predicted or average value of a random variable, calculated by multiplying each possible outcome by its probability and summing the results.

## 2. How is an expected value calculated?

To calculate an expected value, you multiply each possible outcome by its probability, then add all the results together. This is often represented as E(x) = ∑xP(x), where x is the possible outcome and P(x) is its probability.

## 3. Is an expected value the same as the mean?

Yes, the expected value and the mean are essentially the same concept. They both represent the average outcome of a random variable. However, the expected value can also be calculated for non-numerical variables, while the mean only applies to numerical data.

## 4. Can an expected value be negative?

Yes, an expected value can be negative if the possible outcomes of a random variable have negative values and their probabilities are high enough to result in a negative overall expected value. It is important to consider the context and interpretation of a negative expected value in any given situation.

## 5. How is an expected value used in decision making?

An expected value can be used in decision making by comparing the expected values of different options or scenarios. The option with the highest expected value may be the most favorable choice. However, it is important to consider other factors and potential risks in addition to expected value when making decisions.