To calculate the standard deviation of the random variable $X$, follow these steps:

1. Calculate the expected value (mean), $E(X)$: $E(X) = \sum [x \cdot P(X = x)]$ Using the given values:

   $E(X) = (-2)(0.16) + (-1)(0.16) + (0)(0.31) + (1)(0.16) + (2)(0.21)$ $E(X) = -0.32 - 0.16 + 0 + 0.16 + 0.42 = 0.10$

2. Calculate the variance $\text{Var}(X)$: The variance is the expected value of the squared differences from the mean: $\text{Var}(X) = \sum [(x - E(X))^2 \cdot P(X = x)]$ First, calculate $(x - E(X))^2$ for each value of $x$: $(-2 - 0.10)^2 = 4.41, \quad (-1 - 0.10)^2 = 1.21, \quad (0 - 0.10)^2 = 0.01$ $(1 - 0.10)^2 = 0.81, \quad (2 - 0.10)^2 = 3.61$ Now, calculate the weighted sum: $\text{Var}(X) = (4.41)(0.16) + (1.21)(0.16) + (0.01)(0.31) + (0.81)(0.16) + (3.61)(0.21)$ $\text{Var}(X) = 0.7056 + 0.1936 + 0.0031 + 0.1296 + 0.7581 = 1.79$

3. Calculate the standard deviation $\sigma_X$: $\sigma_X = \sqrt{\text{Var}(X)} = \sqrt{1.79} \approx 1.338$

The standard deviation of $X$ is approximately 1.3380 (rounded to 4 decimal places).

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:

1. How do you interpret the standard deviation in a probability distribution?
2. What are other ways to calculate variance and standard deviation in larger datasets?
3. How does changing one probability affect the standard deviation?
4. How is the expected value used in risk assessment in financial markets?
5. What is the difference between population and sample standard deviation?

Tip: When calculating variance, always ensure you subtract the mean from each value before squaring.