What is the expected mean?

The expected mean, also known as the expected value, is a measure of the central tendency of a probability distribution. It represents the average value that we would expect to occur if we were to repeat an experiment an infinite number of times. It is calculated by multiplying each possible outcome by its probability of occurring and summing up all these values.

The expected mean of a random variable is the sum of the product of each possible value of the variable and the probability of that value occurring. It is written as:

E(X) = ∑[x - P(x)]

where:

E(X) is the expected mean of the random variable X.
x is each possible value of the random variable X.
P(x) is the probability of the random variable X taking the value x.

For example, if a random variable X has the following possible values and probabilities:

x P(x)
1 0.2
2 0.5
3 0.3

The expected mean of X would be:

E(X) = 1 - 0.2 + 2 - 0.5 + 3 - 0.3 = 2.1

The expected mean is often interpreted as the "average" value of the random variable. However, it is important to note that the expected mean is not always equal to the actual mean of the random variable. The actual mean is the average value of the random variable when it is observed many times. The expected mean is the average value of the random variable that is expected to be observed in the long run.