To find the expected value $E(x)$ from the table, you multiply each value of $x$ by its corresponding probability $P(x)$ and then sum up the results. The formula is:

$E(x) = \sum (x \times P(x))$

From the table:

$E(x) = 2(0.1) + 4(0.3) + 6(0.4) + 8(0.2)$

The values of $x \times P(x)$ are already provided:

$E(x) = 0.2 + 1.2 + 2.4 + 1.6$

Now, summing these up:

$E(x) = 5.4$

So, the expected value $E(x)$ is 5.4.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. What is the variance of this probability distribution?
2. How is the expected value used in real-life decision making?
3. How would the expected value change if one of the probabilities was altered?
4. Can expected value be applied to continuous distributions?
5. What does it mean if the expected value is not a possible outcome in the dataset?

Tip: Always check if the sum of all probabilities $P(x)$ equals 1, as it's a necessary condition for a valid probability distribution.