Questions: The expected value of a discrete random variable represents the mean value of the outcomes.

Solution Steps

The mean (expected value) of a discrete random variable is calculated using the formula:

\[ \text{Mean} = \sum (x_i \times p_i) \]

where \(x_i\) represents the outcomes and \(p_i\) represents the corresponding probabilities. For our example:

\[ \text{Mean} = 1 \times 0.1 + 2 \times 0.2 + 3 \times 0.3 + 4 \times 0.2 + 5 \times 0.2 = 3.2 \]

The variance \(\sigma^2\) is calculated using the formula:

\[ \text{Variance} = \sigma^2 = \sum ((x_i - \text{Mean})^2 \times p_i) \]

Substituting the values:

\[ \text{Variance} = (1 - 3.2)^2 \times 0.1 + (2 - 3.2)^2 \times 0.2 + (3 - 3.2)^2 \times 0.3 + (4 - 3.2)^2 \times 0.2 + (5 - 3.2)^2 \times 0.2 = 1.56 \]

The standard deviation \(\sigma\) is the square root of the variance:

\[ \text{Standard Deviation} = \sigma = \sqrt{\sigma^2} = \sqrt{1.56} \approx 1.249 \]

Final Answer