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

1. Find the expected value (mean) $E(X)$:

The expected value is calculated by the formula:

$E(X) = \sum [x \cdot P(x)]$

Substituting the given values:

$E(X) = (0 \cdot 0.1) + (1 \cdot 0.15) + (2 \cdot 0.2) + (3 \cdot 0.55)$

$E(X) = 0 + 0.15 + 0.4 + 1.65 = 2.2$

So, the expected value $E(X) = 2.2$.

2. Find the variance $\sigma^2$:

The variance is calculated using the formula:

$\sigma^2 = \sum [ (x - E(X))^2 \cdot P(x) ]$

Now, we calculate the squared deviations from the mean for each value of $x$:

$\sigma^2 = (0 - 2.2)^2 \cdot 0.1 + (1 - 2.2)^2 \cdot 0.15 + (2 - 2.2)^2 \cdot 0.2 + (3 - 2.2)^2 \cdot 0.55$

$\sigma^2 = (-2.2)^2 \cdot 0.1 + (-1.2)^2 \cdot 0.15 + (-0.2)^2 \cdot 0.2 + (0.8)^2 \cdot 0.55$

$\sigma^2 = 4.84 \cdot 0.1 + 1.44 \cdot 0.15 + 0.04 \cdot 0.2 + 0.64 \cdot 0.55$

$\sigma^2 = 0.484 + 0.216 + 0.008 + 0.352$

$\sigma^2 = 1.06$

3. Find the standard deviation $\sigma$:

The standard deviation is the square root of the variance:

$\sigma = \sqrt{1.06} \approx 1.0296$

Thus, the standard deviation is approximately 1.0296.

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Let me know if you'd like more details, or if you have further questions!

Here are some related questions you might find useful:

1. What is the formula for the variance of a probability distribution?
2. How do you interpret the standard deviation in the context of probability distributions?
3. How does the shape of the distribution affect the standard deviation?
4. How would you calculate the mean if a probability distribution changes?
5. What is the difference between the population variance and sample variance?

Tip: When calculating the standard deviation, it is helpful to remember that the variance measures the "spread" of a distribution, and the standard deviation gives that spread in the same units as the data, making it easier to interpret.