Question

consider the probability distribution for the random variable x shown here. complete parts a through c below
x | 10 | 20 | 30 | 40 | 50 | 60
p(x) | 0.05 | 0.25 | 0.30 | 0.20 | 0.10 | 0.10

a. calculate \\( \mu \\), \\( \sigma^2 \\), and \\( \sigma \\).
\\( \mu = \square \\) (type an integer, or a decimal. do not round.)

Explanation:

Step1: Recall the formula for the mean (μ) of a discrete random variable

The formula for the mean \( \mu \) of a discrete random variable \( x \) with probability distribution \( p(x) \) is \( \mu = \sum x \cdot p(x) \).

Step2: Calculate each term \( x \cdot p(x) \)

• For \( x = 10 \) and \( p(x) = 0.05 \): \( 10 \times 0.05 = 0.5 \)
• For \( x = 20 \) and \( p(x) = 0.25 \): \( 20 \times 0.25 = 5 \)
• For \( x = 30 \) and \( p(x) = 0.30 \): \( 30 \times 0.30 = 9 \)
• For \( x = 40 \) and \( p(x) = 0.20 \): \( 40 \times 0.20 = 8 \)
• For \( x = 50 \) and \( p(x) = 0.10 \): \( 50 \times 0.10 = 5 \)
• For \( x = 60 \) and \( p(x) = 0.10 \): \( 60 \times 0.10 = 6 \)

Step3: Sum the terms to find \( \mu \)

\( \mu = 0.5 + 5 + 9 + 8 + 5 + 6 \)
\( \mu = 33.5 \)

Answer:

\( 33.5 \)