Questions: Compute the mean and standard deviation of the random variable with the given discrete probability distribution. x P(x) -4 0.24 -1 0.14 6 0.27 8 0.26 9 0.09 Part 1 of 2 (a) Find the mean. Round the answer to three decimal places, if necessary, The mean is .

Solution Steps

To find the mean \( \mu \) of the discrete random variable, we use the formula:

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

Substituting the values from the distribution:

\[ \mu = (-4 \times 0.24) + (-1 \times 0.14) + (6 \times 0.27) + (8 \times 0.26) + (9 \times 0.09 \]

Calculating each term:

\[ \mu = -0.96 - 0.14 + 1.62 + 2.08 + 0.81 = 3.41 \]

Thus, the mean is \( \mu = 3.41 \).

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

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

Substituting the mean and the values:

\[ \sigma^2 = (-4 - 3.41)^2 \times 0.24 + (-1 - 3.41)^2 \times 0.14 + (6 - 3.41)^2 \times 0.27 + (8 - 3.41)^2 \times 0.26 + (9 - 3.41)^2 \times 0.09 \]

Calculating each term:

\[ \sigma^2 = (-7.41)^2 \times 0.24 + (-4.41)^2 \times 0.14 + (2.59)^2 \times 0.27 + (4.59)^2 \times 0.26 + (5.59)^2 \times 0.09 \]

\[ = 54.9281 \times 0.24 + 19.4481 \times 0.14 + 6.7281 \times 0.27 + 20.0681 \times 0.26 + 31.3681 \times 0.09 \]

Calculating the contributions:

\[ = 13.189 \, + \, 2.7267 \, + \, 1.8146 \, + \, 5.216 \, + \, 2.8231 = 26.002 \]

Thus, the variance is \( \sigma^2 = 26.002 \).

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{26.002} \approx 5.099 \]

Final Answer