0

. 011.011

1

. 033.033

2

. 572.572

3

. 279.279

4

. 083.083

5

. 018.018

6

. 003.003  Compute and interpret the mean of the random variable X.

The numbers given appear to represent a probability distribution, where the decimal values correspond to the probabilities of outcomes indexed from 0 to 6. To compute the mean (expected value) of the random variable \( X \), we need to use the following formula for the expected value \( E(X) \):

\[
E(X) = \sum_{i=0}^{6} x_i p_i
\]

Where:
- \( x_i \) represents the possible values of the random variable \( X \) (i.e., 0, 1, 2, 3, 4, 5, 6),
- \( p_i \) represents the corresponding probabilities (i.e., \( 0.011, 0.033, 0.572, 0.279, 0.083, 0.018, 0.003 \)).

Now let's calculate:

\[
E(X) = (0 \times 0.011) + (1 \times 0.033) + (2 \times 0.572) + (3 \times 0.279) + (4 \times 0.083) + (5 \times 0.018) + (6 \times 0.003)
\]

Breaking down the calculation:

\[
E(X) = 0 + 0.033 + 1.144 + 0.837 + 0.332 + 0.09 + 0.018
\]

\[
E(X) = 2.454
\]

### Interpretation:
The mean (expected value) of the random variable \( X \) is approximately 2.454. This means that, on average, the value of the random variable \( X \) will be around 2.454 when considering the probabilities provided.

Would you like a detailed breakdown of the steps or have any questions?

Here are 5 related questions:
1. How is the variance of a random variable computed from a probability distribution?
2. What does the standard deviation of a random variable tell us?
3. How is the mean different from the mode in a probability distribution?
4. How can we calculate the probability of specific events given this distribution?
5. How would the interpretation of the mean change if the values of \( X \) were shifted?

**Tip:** The mean gives us a central tendency of the data, but it doesn’t provide information about how spread out the values are. For that, you'd need to look at variance or standard deviation.

Compute and interpret the mean of the random variable X with values: 0 (0.011), 1 (0.033), 2 (0.572), 3 (0.279), 4 (0.083), 5 (0.018), 6 (0.003).

Learn how to compute and interpret the mean of a random variable using probabilities: 0 (0.011), 1 (0.033), 2 (0.572), 3 (0.279), 4 (0.083), 5 (0.018), and 6 (0.003). This problem covers key statistical concepts, including expected value calculations. Suitable for students in Grades 10-12.

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