Question

The expected value of a random variable, x, is also called the mean of the distribution of that random variable. Why

do you think it is called the mean?

Answer 1

See explanation below.

Explanation:

If our random variable X is discrete the expected value is given by:

`E(X) = \mu = \sum_(i=1)^n X_i P(X_i)`

Where
`X_i` represent the possible values for the random variable and P the respective probabilities, so then is like a weighted average. The only difference is that the mean is defined as:

`\bar X = (\sum_(i=1)^n X_i)/(n)`

On this mean the weight for each observation is
`(1)/(n)` and for the expected value are different. But the formulas are equivalent.

If our random variable is continuous then the expected value is given by:

`E(X) =\mu = \int_(a)^b f(x) dx`

Where
`f(x)` represent the density function for the random variable and a is the lower limit and b the upper limit where the random variable is defined.

And again is analogous to the mean since we are finding the area below the curve of a function.

We assume that is called mean because is a measure of central tendency in order to see where we have the first moment of a random variable. And since takes in count all the weigths for the possible values for the random variable makes sense called mean.