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

$E(X) = \sum_{i=1}^n x_i p_i$

Where:

• $x_i$ represents the values of $X$,
• $p_i$ represents the probabilities associated with each $x_i$.

Here, the given values are:

$x_1 = -5, \quad x_2 = 2, \quad x_3 = 5$ $p_1 = 0.4, \quad p_2 = 0.4, \quad p_3 = 0.2$

Step 1: Multiply each $x_i$ by its corresponding $p_i$:

$-5 \times 0.4 = -2.0$ $2 \times 0.4 = 0.8$ $5 \times 0.2 = 1.0$

Step 2: Add these results together:

$E(X) = -2.0 + 0.8 + 1.0 = -0.2$

Final Answer:

$E(X) = -0.2$

Would you like a deeper explanation or additional calculations?

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Follow-up Questions:

1. What is the significance of the expected value in probability?
2. How would the expected value change if the probabilities were uniformly distributed?
3. What happens if one of the probabilities $p_i$ is zero?
4. Can you calculate the variance of $X$ for the given distribution?
5. How is the expected value used in real-world applications like finance or decision-making?

Tip:

Ensure all probabilities $p_i$ sum to 1 before calculating $E(X)$, as this is a key property of valid probability distributions.