Questions: HW 5.1 Score: 3 / 16 Answered: 3 / 8 Question 4 Find the mean of the following probability distribution? x P(x) 0 0.2609 1 0.1881 2 0.0474 3 0.3111 4 0.1926 mean = (report answer rounded to one decimal place)

Solution Steps

To find the mean of the probability distribution, we use the formula:

\[ \text{Mean} = \sum (x \times P(x)) \]

Substituting the values from the distribution:

\[ \text{Mean} = 0 \times 0.2609 + 1 \times 0.1881 + 2 \times 0.0474 + 3 \times 0.3111 + 4 \times 0.1926 \]

Calculating this gives:

\[ \text{Mean} = 0 + 0.1881 + 0.0948 + 0.9333 + 0.7704 = 2.0 \]

The variance is calculated using the formula:

\[ \text{Variance} = \sigma^2 = \sum (x - \text{Mean})^2 \times P(x) \]

Substituting the mean and the values:

\[ \text{Variance} = (0 - 2.0)^2 \times 0.2609 + (1 - 2.0)^2 \times 0.1881 + (2 - 2.0)^2 \times 0.0474 + (3 - 2.0)^2 \times 0.3111 + (4 - 2.0)^2 \times 0.1926 \]

Calculating this gives:

\[ \text{Variance} = (4.0) \times 0.2609 + (1.0) \times 0.1881 + (0) \times 0.0474 + (1.0) \times 0.3111 + (4.0) \times 0.1926 \]

\[ \text{Variance} = 1.0436 + 0.1881 + 0 + 0.3111 + 0.7704 = 2.3132 \approx 2.3 \]

The standard deviation is the square root of the variance:

\[ \text{Standard Deviation} = \sigma = \sqrt{\text{Variance}} = \sqrt{2.3} \approx 1.5 \]

Final Answer