Questions: Find the expected value E(X) of the following data. Round your answer to one decimal place. x -6 -5 -4 -3 -2 P(X=x) 0.3 0.1 0.1 0.1 0.4

Solution Steps

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

\[ E(X) = \sum (x \cdot P(X = x)) \]

For the given data:

\[ E(X) = (-6 \times 0.3) + (-5 \times 0.1) + (-4 \times 0.1) + (-3 \times 0.1) + (-2 \times 0.4) \]

Calculating each term:

\[ E(X) = -1.8 - 0.5 - 0.4 - 0.3 - 0.8 = -3.8 \]

Thus, the expected value is:

\[ E(X) = -3.8 \]

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

\[ \sigma^2 = \sum ((x - E(X))^2 \cdot P(X = x)) \]

Substituting \( E(X) = -3.8 \):

\[ \sigma^2 = (-6 - -3.8)^2 \times 0.3 + (-5 - -3.8)^2 \times 0.1 + (-4 - -3.8)^2 \times 0.1 + (-3 - -3.8)^2 \times 0.1 + (-2 - -3.8)^2 \times 0.4 \]

Calculating each term:

\[ \sigma^2 = (2.2)^2 \times 0.3 + (1.2)^2 \times 0.1 + (0.2)^2 \times 0.1 + (0.8)^2 \times 0.1 + (1.8)^2 \times 0.4 \]

\[ = 4.84 \times 0.3 + 1.44 \times 0.1 + 0.04 \times 0.1 + 0.64 \times 0.1 + 3.24 \times 0.4 \]

\[ = 1.452 + 0.144 + 0.004 + 0.064 + 1.296 = 3.0 \]

Thus, the variance is:

\[ \sigma^2 = 3.0 \]

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{3.0} \approx 1.732 \]

Rounding to one decimal place gives:

\[ \sigma \approx 1.7 \]

Final Answer