Questions: A baseball player is asked to swing at pitches in sets of four. The player swings at 100 sets of 4 pitches. The probability distribution for making a particular number of hits is given below, Determine the mean for this discrete probability distribution. x 0 1 2 3 4 P(x) 0.02 0.07 0.22 0.27 0.42 A. 4 B. 3 C. 2 D. 3.5

Solution Steps

The probability distribution for the number of hits \( x \) made by the baseball player in a set of four pitches is given as follows:

\[ \begin{array}{c|c} x & P(x) \\ \hline 0 & 0.02 \\ 1 & 0.07 \\ 2 & 0.22 \\ 3 & 0.27 \\ 4 & 0.42 \\ \end{array} \]

The mean \( \mu \) of the distribution is calculated using the formula:

\[ \mu = \sum_{i=0}^{n} x_i \cdot P(x_i) \]

Substituting the values:

\[ \mu = 0 \times 0.02 + 1 \times 0.07 + 2 \times 0.22 + 3 \times 0.27 + 4 \times 0.42 \]

Calculating each term:

\[ \mu = 0 + 0.07 + 0.44 + 0.81 + 1.68 = 3.0 \]

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

\[ \sigma^2 = \sum_{i=0}^{n} (x_i - \mu)^2 \cdot P(x_i) \]

Substituting the values:

\[ \sigma^2 = (0 - 3.0)^2 \times 0.02 + (1 - 3.0)^2 \times 0.07 + (2 - 3.0)^2 \times 0.22 + (3 - 3.0)^2 \times 0.27 + (4 - 3.0)^2 \times 0.42 \]

Calculating each term:

\[ \sigma^2 = (9) \times 0.02 + (4) \times 0.07 + (1) \times 0.22 + (0) \times 0.27 + (1) \times 0.42 \] \[ \sigma^2 = 0.18 + 0.28 + 0.22 + 0 + 0.42 = 1.1 \]

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{1.1} \approx 1.049 \]

Final Answer