# What is the expected value of a constant variable?

May 11, 2018

The constant itself.

#### Explanation:

If $X$ is a random variable which can only assume the value $x$, then you have

$\setminus m a t h \boldsymbol{E} \left(X\right) = \setminus \mu = x$

This makes sense, since $X$ assumes only one value, and so "on average" it assumes that value as well. Think for example that $X$ equals constantly $3$. Then you sample, for example, ten values from $X$, and you will have ${x}_{1} = 3$, ${x}_{2} = 3 , \ldots , {x}_{10} = 3$, since you can't have anything but $3$. Now, compute the average:

$\setminus \mu = \setminus \frac{{x}_{1} + \ldots + {x}_{10}}{10} = \setminus \frac{3 + \ldots + 3}{10} = \setminus \frac{10 \setminus \cdot 3}{10} = 3$

To be more precise, you may use the definition

$\setminus m a t h \boldsymbol{E} \left(X\right) = \setminus \sum {p}_{i} {x}_{i}$

i.e. the weighted sum of all possible values, weighted with their probabilities. Since $X$ only assumes the value $x$ with probability $1$, you have

$\setminus m a t h \boldsymbol{E} \left(X\right) = \setminus \sum {p}_{i} {x}_{i} = 1 \setminus \cdot x = x$