To find the standard deviation for the given probability distribution, we need to follow these steps:

Step 1: Find the mean (expected value), $\mu$.

The formula for the mean $\mu$ is:

$\mu = \sum (x \cdot P(X = x))$

Where $x$ represents each possible value of $X$, and $P(X = x)$ is the corresponding probability.

Using the data:

$\mu = (-5)(0.2) + (-4)(0.3) + (-3)(0.2) + (-2)(0.1) + (-1)(0.2)$

Calculating each term:

$\mu = (-1) + (-1.2) + (-0.6) + (-0.2) + (-0.2) = -3.2$

So, the mean $\mu = -3.2$.

Step 2: Find the variance.

Variance is given by the formula:

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

Now, we will compute $(x - \mu)^2$ for each $x$ value, then multiply each by its respective probability.

1. For $x = -5$: $(x - \mu)^2 = (-5 + 3.2)^2 = (-1.8)^2 = 3.24$ $P(X = -5) \cdot (x - \mu)^2 = 0.2 \cdot 3.24 = 0.648$

2. For $x = -4$: $(x - \mu)^2 = (-4 + 3.2)^2 = (-0.8)^2 = 0.64$ $P(X = -4) \cdot (x - \mu)^2 = 0.3 \cdot 0.64 = 0.192$

3. For $x = -3$: $(x - \mu)^2 = (-3 + 3.2)^2 = (0.2)^2 = 0.04$ $P(X = -3) \cdot (x - \mu)^2 = 0.2 \cdot 0.04 = 0.008$

4. For $x = -2$: $(x - \mu)^2 = (-2 + 3.2)^2 = (1.2)^2 = 1.44$ $P(X = -2) \cdot (x - \mu)^2 = 0.1 \cdot 1.44 = 0.144$

5. For $x = -1$: $(x - \mu)^2 = (-1 + 3.2)^2 = (2.2)^2 = 4.84$ $P(X = -1) \cdot (x - \mu)^2 = 0.2 \cdot 4.84 = 0.968$

Now, summing these values:

$\sigma^2 = 0.648 + 0.192 + 0.008 + 0.144 + 0.968 = 1.96$

Step 3: Find the standard deviation.

The standard deviation $\sigma$ is the square root of the variance:

$\sigma = \sqrt{1.96} = 1.4$

So, the standard deviation is 1.4 (rounded to one decimal place).

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Would you like further clarification on any of these steps or any related questions? Here's a few more to explore:

1. How would you calculate the mean of a continuous probability distribution?
2. What is the relationship between variance and standard deviation?
3. How can you use standard deviation to understand data spread in a real-life scenario?
4. What happens if a probability distribution is symmetric?
5. Can you use this same method to calculate the variance and standard deviation of a normal distribution?

Tip: To better understand probability distributions, visualize them with a graph—this helps to intuitively grasp how the data is spread out.