Question

A random variable X has the following probability distribution:

X	-2	-1	0	1	2
P(X)	0.2	0.1	0.3	0.2	0.2

The variance of X will be:

• A. 0.1
• B. 1.42
• C. 1.89
• D. 2.54

Hint:

When calculating the variance of a random variable, it's important to first find the expected value \( E(X) \), then calculate \( E(X^2) \) and subtract the square of the expected value. The variance measures the spread or dispersion of a set of values. A higher variance indicates that the values are more spread out from the mean, while a lower variance indicates they are closer to the mean. In probability and statistics, understanding variance is key to analyzing the variability of data.

Answer 1

The Correct Option is C

Variance is calculated as:

\[ \text{Var}(X) = E(X^2) - [E(X)]^2. \]

First, calculate \( E(X) \):

\[ E(X) = \sum X \cdot P(X) = (-2)(0.2) + (-1)(0.1) + (0)(0.3) + (1)(0.2) + (2)(0.2). \]

\[ E(X) = -0.4 - 0.1 + 0 + 0.2 + 0.4 = 0.1. \]

Next, calculate \( E(X^2) \):

\[ E(X^2) = \sum X^2 \cdot P(X) = (-2)^2(0.2) + (-1)^2(0.1) + (0)^2(0.3) + (1)^2(0.2) + (2)^2(0.2). \]

\[ E(X^2) = 4(0.2) + 1(0.1) + 0(0.3) + 1(0.2) + 4(0.2) = 0.8 + 0.1 + 0 + 0.2 + 0.8 = 1.9. \]

Finally, calculate the variance:

\[ \text{Var}(X) = E(X^2) - [E(X)]^2 = 1.9 - (0.1)^2 = 1.9 - 0.01 = 1.89. \]

Answer 2

Variance is calculated as:

\[ \text{Var}(X) = E(X^2) - [E(X)]^2. \]

Step 1: Calculate \( E(X) \):

The expected value \( E(X) \) is calculated using the formula: \[ E(X) = \sum X \cdot P(X). \] Substituting the given values: \[ E(X) = (-2)(0.2) + (-1)(0.1) + (0)(0.3) + (1)(0.2) + (2)(0.2). \] Simplifying the terms: \[ E(X) = -0.4 - 0.1 + 0 + 0.2 + 0.4 = 0.1. \] Therefore, \( E(X) = 0.1 \).

Step 2: Calculate \( E(X^2) \):

The expected value of \( X^2 \), \( E(X^2) \), is calculated as: \[ E(X^2) = \sum X^2 \cdot P(X). \] Substituting the given values: \[ E(X^2) = (-2)^2(0.2) + (-1)^2(0.1) + (0)^2(0.3) + (1)^2(0.2) + (2)^2(0.2). \] Simplifying the terms: \[ E(X^2) = 4(0.2) + 1(0.1) + 0(0.3) + 1(0.2) + 4(0.2) = 0.8 + 0.1 + 0 + 0.2 + 0.8 = 1.9. \] Therefore, \( E(X^2) = 1.9 \).

Step 3: Calculate the variance:

Finally, the variance is calculated as: \[ \text{Var}(X) = E(X^2) - [E(X)]^2 = 1.9 - (0.1)^2 = 1.9 - 0.01 = 1.89. \]

Conclusion: The variance \( \text{Var}(X) = 1.89 \).