I am given a table of the lifetimes of a set of components measured to the nearest 100 hours. One column has lifetime (in hours) and the other has frequency. I am then told to calculate the mean lifetime in hours.
I understand that the expected value is the the same as the mean, and the formula for expected value is $\sum_{range(X)} xp_X(x)$ where $X$ is a discrete random variable and $p_X(x)$ is the probability mass function.
For this problem, It seems to me that $x =$ lifetime and $p_X(x) =$ frequency.
Applying the formula for the expected value to the two columns, I get $\sum_{range(X)} xp_X(x) = 26,300$.
Here's my confusion: The expected value formula states that this should be the mean, but the problem itself then divides $26,300$ by the sum of the frequencies, which is $68$. Therefore, the solution should apparently be $\dfrac{26,300}{68} = 387$.
So why did the expected value formula not give us the mean then? Why are we required to then divide by $68$? I was under the impression that, in order to find the mean, all I had to do was apply the expected value formula?
I would greatly appreciate it if people could please take the time to clarify this for me.
The mean of some number of realizations of a random variable is in general different from the expected value of the random variable.
Mean: Sum of lifetimes divided by number of objects in sample.
Expected value: What you've described.
For example, say $X_1,X_2$ are flips of a fair coin, where $X_1=1$ when the first coin is heads, and $X_1=0$ when it is tails - same for the second coin. Then $E(X_1+X_2)=1$ is the expected number of heads, but $X_1+X_2$ can take values in $\{0,1,2\}$, depending on how the coins turn up.
One way to look at it: Mean is a "sample" notion, and expected value is a "population" notion.