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To find the standard deviation of a probability distribution, we follow these steps:

### Step 1: Find the mean \( \mu \)
The mean \( \mu \) is given by:
\[
\mu = \sum x \cdot P(x)
\]
Where \( x \) is each value of the random variable and \( P(x) \) is the corresponding probability.

The data provided is:

| \( x \) | \( P(x) \) |
|--------|------------|
| 0      | 0.1        |
| 1      | 0.15       |
| 2      | 0.15       |
| 3      | 0.6        |

### Step 2: Find the variance
The variance \( \sigma^2 \) is calculated by:
\[
\sigma^2 = \sum P(x) \cdot (x - \mu)^2
\]

### Step 3: Find the standard deviation
The standard deviation \( \sigma \) is the square root of the variance:
\[
\sigma = \sqrt{\sigma^2}
\]

Let me calculate the values step by step.The calculations are as follows:

- The mean \( \mu \) is \( 2.25 \).
- The variance \( \sigma^2 \) is \( 1.0875 \).
- The standard deviation \( \sigma \) is approximately \( 1.04 \) (rounded to two decimal places).

If you have any questions or want further details, feel free to ask!

Here are some related questions to explore:
1. How is variance related to standard deviation?
2. Why do we square the difference from the mean when calculating variance?
3. What does the standard deviation represent in a probability distribution?
4. How does increasing probabilities affect the mean and standard deviation?
5. Can a standard deviation ever be negative?

**Tip**: Standard deviation provides insight into how spread out the values in a dataset are, relative to the mean.

Find the standard deviation of this probability distribution. Give your answer to at least 2 decimal places.

Calculate the standard deviation of a probability distribution using the formula for mean and variance. Given probabilities and values for x, we find the mean (μ), variance (σ²), and standard deviation (σ). Suitable for high school students learning statistics.

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