The expected value of a random variable $X$, denoted as $E(X)$, is calculated using the formula:

$E(X) = \sum_{i} x_i P(X = x_i)$

Where $x_i$ are the possible values of $X$, and $P(X = x_i)$ is the corresponding probability.

Using the given data:

$x = \{-30, -10, 0, 10, 20, 40\}$ $P(X = x) = \{0.1, 0.1, 0.1, 0.5, 0, 0.2\}$

We calculate $E(X)$:

$E(X) = (-30)(0.1) + (-10)(0.1) + (0)(0.1) + (10)(0.5) + (20)(0) + (40)(0.2)$

Performing the calculations step-by-step:

1. $(-30)(0.1) = -3$
2. $(-10)(0.1) = -1$
3. $(0)(0.1) = 0$
4. $(10)(0.5) = 5$
5. $(20)(0) = 0$
6. $(40)(0.2) = 8$

Summing these:

$E(X) = -3 - 1 + 0 + 5 + 0 + 8 = 9$

Final Answer:

$E(X) = 9$

Would you like further clarification or related insights? Here are some related questions to explore:

1. How does the expected value change with modified probabilities?
2. What is the variance of $X$ for this distribution?
3. How would the expected value change if $x = 20$ had a probability of 0.2?
4. Can expected value be used to predict outcomes in the long run?
5. What happens to $E(X)$ if all probabilities are doubled?

Tip: Expected value provides the mean outcome of a probability distribution, which is useful for understanding the "average" result in probabilistic scenarios.