Questions: Compute the mean and standard deviation of the random variable with the given discrete probability distribution. x P(x) -2 0.1 2 0.26 3 0.28 4 0.14 7 0.22 Part 1 of 2 (a) Find the mean. Round the answer to three decimal places, if necessary. The mean is . Part 2 of 2 (b) Find the standard deviation. Round the answer to three decimal places, if necessary. The standard deviation 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:

\[ \mu = (-2 \times 0.1) + (2 \times 0.26) + (3 \times 0.28) + (4 \times 0.14) + (7 \times 0.22 \]

Calculating each term:

\[ \mu = -0.2 + 0.52 + 0.84 + 0.56 + 1.54 = 3.26 \]

Thus, the mean is:

\[ \boxed{\mu = 3.26} \]

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

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

Substituting the mean and the values:

\[ \sigma^2 = (-2 - 3.26)^2 \times 0.1 + (2 - 3.26)^2 \times 0.26 + (3 - 3.26)^2 \times 0.28 + (4 - 3.26)^2 \times 0.14 + (7 - 3.26)^2 \times 0.22 \]

Calculating each term:

\[ \sigma^2 = (-5.26)^2 \times 0.1 + (-1.26)^2 \times 0.26 + (-0.26)^2 \times 0.28 + (0.74)^2 \times 0.14 + (3.74)^2 \times 0.22 \]

\[ = 27.6676 \times 0.1 + 1.5876 \times 0.26 + 0.0676 \times 0.28 + 0.5476 \times 0.14 + 13.9876 \times 0.22 \]

Calculating the contributions:

\[ = 2.76676 + 0.413376 + 0.018928 + 0.076664 + 3.077232 = 6.352 \]

Thus, the variance is:

\[ \boxed{\sigma^2 = 6.352} \]

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{6.352} \approx 2.52 \]

Thus, the standard deviation is:

\[ \boxed{\sigma = 2.52} \]

Final Answer