Question

a project is graded on a scale of 1 to 5. if the random variable, x, is the project grade, what is the mean of the probability distribution below? probability distribution bar graph with x - axis values 1, 2, 3, 4, 5 and y - axis (probability) with markings 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4 options: 0.2, 0.4, 1, 3 (approximate, based on image)

Explanation:

Step1: Identify probabilities for each \( X \)

From the graph:

• \( P(X = 1) = 0.1 \)
• \( P(X = 2) = 0.2 \)
• \( P(X = 3) = 0.4 \)
• \( P(X = 4) = 0.2 \)
• \( P(X = 5) = 0.1 \)

Step2: Use the mean formula for a discrete random variable

The formula for the mean \( \mu \) of a discrete random variable is \( \mu = \sum [x \cdot P(x)] \).

Step3: Calculate each term \( x \cdot P(x) \)

• For \( x = 1 \): \( 1 \cdot 0.1 = 0.1 \)
• For \( x = 2 \): \( 2 \cdot 0.2 = 0.4 \)
• For \( x = 3 \): \( 3 \cdot 0.4 = 1.2 \)
• For \( x = 4 \): \( 4 \cdot 0.2 = 0.8 \)
• For \( x = 5 \): \( 5 \cdot 0.1 = 0.5 \)

Step4: Sum the terms

\( \mu = 0.1 + 0.4 + 1.2 + 0.8 + 0.5 = 3 \)

Answer:

3