Calculate the expected value for the following probability distribution:
x     p (x) 
1       .15
2      .35
3      .40
4      .10

To calculate the expected value \(E(X)\) of a probability distribution, you use the following formula:

\[
E(X) = \sum [x \cdot p(x)]
\]

In this case, we have the following values for \(x\) and \(p(x)\):

\[
E(X) = (1 \cdot 0.15) + (2 \cdot 0.35) + (3 \cdot 0.40) + (4 \cdot 0.10)
\]

Now, let's compute each term:

\[
E(X) = (1 \cdot 0.15) + (2 \cdot 0.35) + (3 \cdot 0.40) + (4 \cdot 0.10)
\]
\[
E(X) = 0.15 + 0.70 + 1.20 + 0.40
\]
\[
E(X) = 2.45
\]

Thus, the expected value \(E(X)\) is \(2.45\).

Would you like to see more details or clarifications on this?

Here are 5 related questions:
1. How is variance calculated for a probability distribution?
2. Can expected value be a value that is not in the original data set?
3. What is the meaning of expected value in a real-world context?
4. How does the expected value change if the probabilities are adjusted?
5. What is the role of expected value in decision theory?

**Tip**: The expected value is essentially a weighted average, with the probabilities acting as weights for the different outcomes.

Calculate the expected value for the following probability distribution: x p(x) 1 0.15 2 0.35 3 0.40 4 0.10

Learn how to calculate the expected value for a given probability distribution with x values of 1, 2, 3, and 4, and corresponding probabilities. This problem covers key concepts in probability, including the expected value formula and how to compute a weighted average.

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