Question

find the mean (expected value) of the probability distribution.

x	p(x)
80	.23
85	.42
90	.11
95	.12

options: 83.2, 87.1, 85, 84.4

Explanation:

Step1: Recall the formula for the mean (expected value) of a probability distribution, which is \( \mu = \sum x \cdot P(x) \).

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

• For \( x = 75 \) and \( P(x) = 0.12 \): \( 75 \times 0.12 = 9 \)
• For \( x = 80 \) and \( P(x) = 0.23 \): \( 80 \times 0.23 = 18.4 \)
• For \( x = 85 \) and \( P(x) = 0.42 \) (assuming the typo "A2" is 0.42): \( 85 \times 0.42 = 35.7 \)
• For \( x = 90 \) and \( P(x) = 0.11 \): \( 90 \times 0.11 = 9.9 \)
• For \( x = 95 \) and \( P(x) = 0.12 \): \( 95 \times 0.12 = 11.4 \)

Step3: Sum up all these terms: \( 9 + 18.4 + 35.7 + 9.9 + 11.4 \)

First, \( 9 + 18.4 = 27.4 \)
Then, \( 27.4 + 35.7 = 63.1 \)
Next, \( 63.1 + 9.9 = 73 \)
Finally, \( 73 + 11.4 = 84.4 \)

Answer:

84.4