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

$\text{Variance} (\sigma^2) = \sum (x - \mu)^2 P(X = x)$

where:

• $x$ represents the possible values of the random variable,
• $P(X = x)$ is the probability of each corresponding value of $x$,
• $\mu$ is the expected value (mean) of the distribution.

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

The expected value is given by:

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

Substitute the values:

$\mu = (5 \times 0.3) + (6 \times 0.2) + (7 \times 0.1) + (8 \times 0.2) + (9 \times 0.2)$ $\mu = 1.5 + 1.2 + 0.7 + 1.6 + 1.8 = 7.8$

So, $\mu = 7.8$.

Step 2: Calculate the variance.

Now that we have the mean, we can calculate the variance:

$\sigma^2 = (5 - 7.8)^2 \cdot 0.3 + (6 - 7.8)^2 \cdot 0.2 + (7 - 7.8)^2 \cdot 0.1 + (8 - 7.8)^2 \cdot 0.2 + (9 - 7.8)^2 \cdot 0.2$

Let's break it down:

$(5 - 7.8)^2 = (-2.8)^2 = 7.84$ $(6 - 7.8)^2 = (-1.8)^2 = 3.24$ $(7 - 7.8)^2 = (-0.8)^2 = 0.64$ $(8 - 7.8)^2 = (0.2)^2 = 0.04$ $(9 - 7.8)^2 = (1.2)^2 = 1.44$

Now substitute these values into the variance formula:

$\sigma^2 = (7.84 \times 0.3) + (3.24 \times 0.2) + (0.64 \times 0.1) + (0.04 \times 0.2) + (1.44 \times 0.2)$ $\sigma^2 = 2.352 + 0.648 + 0.064 + 0.008 + 0.288 = 3.36$

Final Answer:

The variance is $\sigma^2 = 3.4$ (rounded to one decimal place).

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Would you like to dive deeper into any step or need further clarification?

Here are some related questions:

1. How do we calculate the standard deviation from the variance?
2. What are the differences between variance and mean absolute deviation (MAD)?
3. How do probability distributions affect the calculation of the mean and variance?
4. What is the significance of the variance in a data set?
5. How does the shape of a distribution affect the variance?

Tip: When calculating the variance, always ensure the probabilities sum to 1. This is a key property of any probability distribution!