To calculate the expected value $E(X)$ of the discrete random variable $X$, we use the formula:

$E(X) = \sum (x_i \cdot P(X = x_i))$

From the image, the values of $X$ and their corresponding probabilities $P(X = x_i)$ are:

• For $X = -5$, $P(X = -5) = 0.2$
• For $X = 0$, $P(X = 0) = 0.4$
• For $X = 5$, $P(X = 5) = 0.4$

Now, let's compute the expected value:

$E(X) = (-5 \cdot 0.2) + (0 \cdot 0.4) + (5 \cdot 0.4)$

$E(X) = (-1) + (0) + (2)$

$E(X) = 1$

So, the expected value $E(X)$ is 1.

Would you like further details or have any questions?

Here are five related questions:

1. How is the expected value different from the median in a probability distribution?
2. What does the expected value represent in practical scenarios?
3. How can the variance of $X$ be computed from this distribution?
4. What happens to the expected value if the probabilities are changed?
5. What is the impact of adding a constant to each $x_i$ on the expected value?

Tip: The expected value is often thought of as the "average" outcome if you could repeat the experiment many times.