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To find the mean of the given probability distribution, you use the formula for the expected value \( E(X) \):

\[
E(X) = \sum (x_i \cdot P(x_i))
\]

Where:
- \( x_i \) are the values of the random variable.
- \( P(x_i) \) are their corresponding probabilities.

From the table in the image:
- \( x_0 = 0 \), \( P(0) = 0.25 \)
- \( x_1 = 1 \), \( P(1) = 0.15 \)
- \( x_2 = 2 \), \( P(2) = 0.1 \)
- \( x_3 = 3 \), \( P(3) = 0.5 \)

Now, we calculate the expected value \( E(X) \).

\[
E(X) = (0 \cdot 0.25) + (1 \cdot 0.15) + (2 \cdot 0.1) + (3 \cdot 0.5)
\]

Let me compute that.The mean (expected value) of this probability distribution is \( 1.85 \). When rounded to one decimal place, the mean is \( 1.9 \).

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What is the variance of this probability distribution?
2. How do you calculate the standard deviation from a probability distribution?
3. What does the mean represent in the context of probability distributions?
4. How can we verify that this is a valid probability distribution?
5. What is the probability that \( X \geq 2 \) based on this distribution?

**Tip:** In probability distributions, the mean represents the long-run average outcome after many trials.

Find the mean of this probability distribution. Round your answer to one decimal place.

Learn how to find the mean (expected value) of a given probability distribution. This problem uses the formula E(X) = Σ (x_i * P(x_i)) and involves values of x = 0, 1, 2, 3 with their respective probabilities. Suitable for students in Grades 9-12.

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