# Is my understanding of Expected Value of a Random Variable correct?

I have come up with the following understanding about Expectation of RV.

Please, correct me if I am wrong.

Definition: Expected value of a RV says that, if a random experiment is repeated n number of times, what would be the value that you expect to see most of the times. Expected value of a Random Variable is a value. It is not a probability.

Example: In case of a dice, if the dice is rolled $n$ number of times, the value we would see most of the times is 3.5.

Calculation: We don't have to repeat the same experiment $n$ number of times to find the expected value of a random variable. It can be done using the random variable and its associated probability distribution. In case of a dice,

$\sum_{x=1}^6 x(1/6) = 1*(1/6) + 2*(1/6) + 3*(1/6) + 4*(1/6) + 5*(1/6) + 6*(1/6) = 3.5$

Continuous Case

If $X$ is a continuous random variable with density function $f_X(x),$ then the expectation of $X$ is $\displaystyle E(X)=\int_{-\infty}^\infty x\,f_X(x)\,dx$

If $Y=g(X)$ is a function of $X$(i.e. a density function)$,$ then $\displaystyle E(Y)=\int_{-\infty}^\infty g(x)\,f_X(x)\,dx$

Edit:

Definition:

Expected value of a RV says that, if a random experiment is repeated n number of times, what would be the average value of the outcomes.

Expected value of a Random Variable is a value. It is not a probability.

Example:

In case of a dice, if the dice is rolled $n$ number of times, the average value of the outcomes would be 3.5.

• This definition is wrong, it defines mode, not EV. Consider a fair coin (0 - tails, 1 - heads), it's expected value is 1/2 * 0 + 1/2 * 1 = 0.5. You will never see value of 0.5 since this variable has only values in {0, 1}. – Tim Oct 6 '16 at 12:58
• @Tim, How can I correct my definition in its current form? – user366312 Oct 6 '16 at 13:01
• I, for one, have never seen a value of 3.5 on a die. – whuber Oct 6 '16 at 13:08
• @whuber, Is my edit correct now? – user366312 Oct 6 '16 at 13:13
• Your edit, which reflects the answer you have received, expresses a derived result: it is a (qualitative) statement of the weak law of large numbers. Although that certainly can help with understanding, it is not a definition of expected value. Your original "example" still expresses the bizarre idea that a die can actually exhibit a value of 3.5! The expected value is a property of a model for your experiment (a die roll), as I have explained in various answers such as the one at stats.stackexchange.com/a/30330/919. – whuber Oct 6 '16 at 13:16

I would like to help - you put $n$ in your definition, but this is exactly what we don't want. Since $n$ is never referenced elsewhere, its presence serves to inform the reader of your definition that the experiment is repeated a finite number of times. So, we can either remove $n$ altogether and state that the experiment is repeated infinitely many times (which is a problem logically) or we can leave $n$ right where it is and state that the expectation is the limit of the average of the values as $n\to\infty$. That is, in your dice example, let's take the average of $n$ values. As we increase $n$, the average gets closer and closer to $3.5$.