To find the variance $\text{Var}(X)$ of a discrete random variable $X$, we need to follow these steps:

Step 1: Calculate the expected value $E(X)$.

The expected value is calculated as: $E(X) = \sum (x_i \cdot P(X = x_i))$ where $x_i$ are the possible values of $X$ and $P(X = x_i)$ are the corresponding probabilities.

From the table: $E(X) = (1 \cdot 0.12) + (2 \cdot 0.24) + (3 \cdot 0.36) + (4 \cdot 0.17) + (5 \cdot 0.11)$ $E(X) = 0.12 + 0.48 + 1.08 + 0.68 + 0.55 = 2.91$

Step 2: Calculate $E(X^2)$.

To find $E(X^2)$, we first square the $x$-values and then multiply by their respective probabilities: $E(X^2) = \sum (x_i^2 \cdot P(X = x_i))$ $E(X^2) = (1^2 \cdot 0.12) + (2^2 \cdot 0.24) + (3^2 \cdot 0.36) + (4^2 \cdot 0.17) + (5^2 \cdot 0.11)$ $E(X^2) = (1 \cdot 0.12) + (4 \cdot 0.24) + (9 \cdot 0.36) + (16 \cdot 0.17) + (25 \cdot 0.11)$ $E(X^2) = 0.12 + 0.96 + 3.24 + 2.72 + 2.75 = 9.79$

Step 3: Calculate the variance $\text{Var}(X)$.

The variance is given by: $\text{Var}(X) = E(X^2) - (E(X))^2$ $\text{Var}(X) = 9.79 - (2.91)^2 = 9.79 - 8.4681 = 1.3219$

So, the variance of $X$ is $\text{Var}(X) = 1.3219$.

Would you like more details on any step, or do you have any questions?

Relative questions:

1. How is the standard deviation related to variance?
2. What are the properties of a probability distribution?
3. How do you find the expected value of a continuous random variable?
4. What is the significance of variance in statistics?
5. Can you explain how the law of large numbers relates to expected value?

Tip:

Remember, variance measures how spread out the values of a random variable are from the mean.