Questions: Let X represent the number of tires with low air pressure on a randomly chosen car. The probability distribution of X is as follows. x 0 1 2 3 4 P(x) 0.2 0.1 0.3 0.3 0.1 Part 1 of 2 (a) Compute the mean μX. Round the answer to one decimal place. μX=

Solution Steps

To compute the mean \( \mu_X \) of the random variable \( X \), we use the formula:

\[ \mu_X = \sum_{x} x \cdot P(x) \]

Substituting the values from the probability distribution:

\[ \mu_X = 0 \times 0.2 + 1 \times 0.1 + 2 \times 0.3 + 3 \times 0.3 + 4 \times 0.1 \]

Calculating each term:

\[ \mu_X = 0 + 0.1 + 0.6 + 0.9 + 0.4 = 2.0 \]

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

\[ \sigma^2 = \sum_{x} (x - \mu_X)^2 \cdot P(x) \]

Substituting the values:

\[ \sigma^2 = (0 - 2.0)^2 \times 0.2 + (1 - 2.0)^2 \times 0.1 + (2 - 2.0)^2 \times 0.3 + (3 - 2.0)^2 \times 0.3 + (4 - 2.0)^2 \times 0.1 \]

Calculating each term:

\[ \sigma^2 = (4.0) \times 0.2 + (1.0) \times 0.1 + (0) \times 0.3 + (1.0) \times 0.3 + (4.0) \times 0.1 \]

\[ \sigma^2 = 0.8 + 0.1 + 0 + 0.3 + 0.4 = 1.6 \]

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{1.6} \approx 1.3 \]

Final Answer