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

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

Where:

• $x$ is each outcome
• $P(x)$ is the probability of that outcome

Step-by-step calculation:

1. Multiply each outcome ($x$) by its corresponding probability ($P(x)$):

   $-5 \cdot 0.2 = -1.0$ $2 \cdot 0.3 = 0.6$ $4 \cdot 0.4 = 1.6$ $7 \cdot 0.1 = 0.7$

2. Add these values together:

   $E(X) = -1.0 + 0.6 + 1.6 + 0.7 = 1.9$

Final Answer:

$\text{Expected Value} = \mathbf{1.90}$

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Would you like a breakdown of the formula or an explanation of its application?

Here are 5 related questions for practice:

1. What is the expected value if probabilities are adjusted slightly?
2. How does the variance of this distribution compare to the mean?
3. How would adding a constant to each outcome affect the expected value?
4. What happens to $E(X)$ if all probabilities are doubled?
5. Can you compute the expected value of a continuous random variable?

Tip: Expected value gives the long-term average of a random variable based on its probabilities!