Question

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

Answer 1

Answer:

See explanation below.

Step-by-step 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:

$\bar X = \frac{\sum_{i=1}^n X_i}{n}$

On this mean the weight for each observation is $\frac{1}{n}$ 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.