To calculate the expected value of a random variable, you use the formula:

$E(X) = \sum_{x} x \cdot P(x)$

Here’s the step-by-step process based on the given graph:

1. Identify the values of $x$ (horizontal axis).
2. Find the corresponding probabilities $P(x)$ (height of the bars on the vertical axis).
3. Multiply each $x$ by its corresponding $P(x)$.
4. Sum the results.

Let me extract the values from the graph in the image:

From the graph:

• $x = 0, 1, 2, 3, 4, 5$
• Probabilities $P(x)$:
  • $P(0) = 0$
  • $P(1) = 0.2$
  • $P(2) = 0.1$
  • $P(3) = 0.1$
  • $P(4) = 0.2$
  • $P(5) = 0.4$

Calculation:

$E(X) = (0 \cdot 0) + (1 \cdot 0.2) + (2 \cdot 0.1) + (3 \cdot 0.1) + (4 \cdot 0.2) + (5 \cdot 0.4)$ $E(X) = 0 + 0.2 + 0.2 + 0.3 + 0.8 + 2.0$ $E(X) = 3.5$

Final Answer:

The expected value is $3.5$.

Let me know if you'd like further details or explanations!

Here are 5 questions to expand on this topic:

1. What does the expected value of a random variable represent in real-world terms?
2. How is the expected value affected if one probability is adjusted?
3. Can the expected value ever be a number not within the range of $x$ values?
4. How does the expected value differ from the median of a distribution?
5. What happens to $E(X)$ if all probabilities are equally distributed?

Tip: Always ensure the sum of all probabilities $\sum P(x)$ equals 1 when solving such problems!