Let's work through the calculations for variance $\sigma^2$ using the information from the table.

Formula Recap:

For a discrete random variable $X$ with probabilities $P(X)$, the variance $\sigma^2$ is given by:

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

where:

• $x_i$ are the values of $X$,
• $P(X = x_i)$ are the corresponding probabilities, and
• $\mu$ is the mean (which is given as $\mu = 12.44$).

Given Data:

The values of $X$ and their respective probabilities $P(X)$ are: $X: 1, 3, 5, 10, 16, 19, 20$ $P(X): 0.14, 0.1, 0.1, 0.12, 0.1, 0.08, 0.36$

Step-by-Step Calculation:

1. Subtract the mean $\mu = 12.44$ from each $x_i$.
2. Square the result of each subtraction.
3. Multiply each squared result by the corresponding probability $P(X)$.
4. Sum all the results to get $\sigma^2$.

Let me calculate that for you.The variance $\sigma^2$ is approximately $58.77$.

If you have any further questions or want a detailed explanation, feel free to ask!

Here are five related questions you might find interesting:

1. How is the standard deviation related to the variance?
2. What is the significance of variance in statistics?
3. Can you explain the difference between population variance and sample variance?
4. How would the variance change if one of the probabilities increased or decreased?
5. How does the shape of the probability distribution relate to the variance?

Tip: Variance gives you an idea of how spread out the values of a random variable are around the mean. The larger the variance, the more spread out the values.