https://file.aiquickdraw.com/mathbot/file/1726860409826t067ndhg.jpg

Let's first analyze whether the table shown corresponds to a valid probability distribution.

A valid probability distribution must meet the following conditions:

1. Each probability \( P(x) \) must be between 0 and 1, inclusive.
2. The sum of all probabilities must be equal to 1.

The table provides the following information:

\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
0 & 0.034 \\
1 & 0.165 \\
2 & 0.301 \\
3 & 0.301 \\
4 & 0.165 \\
5 & 0.034 \\
\hline
\end{array}
\]

### Check 1: Probability between 0 and 1

All given probabilities \( P(x) \) are within the range [0, 1], so this condition is satisfied.

### Check 2: Sum of Probabilities

Let's calculate the sum of all probabilities \( P(x) \):

\[
\text{Sum} = 0.034 + 0.165 + 0.301 + 0.301 + 0.165 + 0.034
\]

I'll calculate this sum.The sum of the probabilities is \( 1.0 \), which satisfies the second condition.

### Conclusion:
Since both conditions are met, the table represents a valid probability distribution. Therefore, the correct answer to the first question is:

- \( \boxed{A.} \) Yes, the table shows a probability distribution.

Now, we will calculate the **mean** and **standard deviation** of the random variable \( x \).

The mean \( \mu \) of a probability distribution is given by:

\[
\mu = \sum x \cdot P(x)
\]

The standard deviation \( \sigma \) is given by:

\[
\sigma = \sqrt{\sum \left(x - \mu \right)^2 \cdot P(x)}
\]

I'll now calculate both the mean and standard deviation.The mean \( \mu \) of the random variable \( x \) is 2.5, and the standard deviation \( \sigma \) is approximately 1.15 (rounded to one decimal place).

Thus, the answers for the remaining parts are:

1. The mean is \( 2.5 \) children.
2. The standard deviation is \( 1.1 \) children (rounded to one decimal place).

Do you have any further questions or would you like more details?

Here are 5 related questions to expand on the concept:

1. How would you interpret the mean in the context of this problem?
2. Why is it important that the sum of probabilities equals 1?
3. What does the standard deviation tell us about the distribution of \( x \)?
4. How would the results change if one of the probabilities was incorrect?
5. Can you construct another valid probability distribution with different values for \( x \)?

**Tip:** The standard deviation provides insight into the spread of the distribution — a larger value indicates more variability around the mean.

Does the table show a probability distribution? If yes, find the mean and standard deviation.

Explore how to determine if a table represents a probability distribution. Learn to calculate the mean and standard deviation for a random variable with detailed formulas. Suitable for Grades 9-12.

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