Questions: Find the expected value of the random variable with the given probability distribution. [ mu=[?] ] Do not round your answer. x P -3 0.10 -1 0.15 1 0.20 3 0.25 5 0.20 7 0.10

Solution Steps

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

\[ \mu = \sum (x \cdot P) \]

Substituting the values from the probability distribution:

\[ \mu = (-3 \times 0.1) + (-1 \times 0.15) + (1 \times 0.2) + (3 \times 0.25) + (5 \times 0.2) + (7 \times 0.1) \]

Calculating each term:

\[ \mu = -0.3 - 0.15 + 0.2 + 0.75 + 1 + 0.7 = 2.2 \]

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

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

Substituting the values:

\[ \sigma^2 = (-3 - 2.2)^2 \times 0.1 + (-1 - 2.2)^2 \times 0.15 + (1 - 2.2)^2 \times 0.2 + (3 - 2.2)^2 \times 0.25 + (5 - 2.2)^2 \times 0.2 + (7 - 2.2)^2 \times 0.1 \]

Calculating each term:

\[ \sigma^2 = (5.2^2 \times 0.1) + (3.2^2 \times 0.15) + (1.2^2 \times 0.2) + (0.8^2 \times 0.25) + (2.8^2 \times 0.2) + (4.8^2 \times 0.1) \]

\[ = (27.04 \times 0.1) + (10.24 \times 0.15) + (1.44 \times 0.2) + (0.64 \times 0.25) + (7.84 \times 0.2) + (23.04 \times 0.1) \]

\[ = 2.704 + 1.536 + 0.288 + 0.16 + 1.568 + 2.304 = 8.56 \]

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{8.56} \approx 2.9258 \]

Final Answer