To find the variance of a discrete random variable $X$, we use the following formula:

$\text{Var}(X) = \sum_{i} P(X = x_i) \cdot (x_i - \mu)^2$

Where:

• $P(X = x_i)$ is the probability associated with each value of $X$,
• $x_i$ are the possible values of $X$,
• $\mu$ is the expected value (mean) of $X$, given by:

$\mu = \sum_{i} P(X = x_i) \cdot x_i$

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

We already have the data:

$x = \{-5, -4, -3, -2, -1\}$ $P(X = x) = \{0.2, 0.3, 0.2, 0.1, 0.2\}$

Now, calculate the expected value $\mu$:

$\mu = (0.2 \times -5) + (0.3 \times -4) + (0.2 \times -3) + (0.1 \times -2) + (0.2 \times -1)$ $\mu = -1.0 - 1.2 - 0.6 - 0.2 - 0.2 = -3.2$

Step 2: Calculate the variance

Now, we calculate the variance:

$\text{Var}(X) = 0.2 \cdot (-5 + 3.2)^2 + 0.3 \cdot (-4 + 3.2)^2 + 0.2 \cdot (-3 + 3.2)^2 + 0.1 \cdot (-2 + 3.2)^2 + 0.2 \cdot (-1 + 3.2)^2$

$= 0.2 \cdot (-1.8)^2 + 0.3 \cdot (-0.8)^2 + 0.2 \cdot (0.2)^2 + 0.1 \cdot (1.2)^2 + 0.2 \cdot (2.2)^2$

Now, calculating each squared term:

$= 0.2 \cdot 3.24 + 0.3 \cdot 0.64 + 0.2 \cdot 0.04 + 0.1 \cdot 1.44 + 0.2 \cdot 4.84$

$= 0.648 + 0.192 + 0.008 + 0.144 + 0.968 = 1.96$

Final Answer:

The variance of $X$ is approximately:

$\boxed{1.96}$

Would you like any further explanation or clarification on the calculation?

Related Questions:

1. How do you calculate the standard deviation from variance?
2. What happens if the probabilities are not valid (sum to more than 1)?
3. How can you calculate the expected value of a continuous random variable?
4. What is the difference between variance and standard deviation?
5. How would you calculate the skewness of a distribution?

Tip: When working with probabilities, always ensure they sum to 1 to make sure your distribution is valid!