Question

Calculate the mean (expected value) of a random variable \( X \) given its probability distribution setup below: 

\( X \)	1	2	3
\( P(X) \)	\( 0.2 \)	\( 0.5 \)	\( 0.3 \)

• A. \( 2.1 \)
• B. \( 2.0 \)
• C. \( 1.0 \)
• D. \( 3.0 \)

Hint:

The expected value does not have to be a whole number, nor does it have to match one of the exact coordinates listed in your table. It simply represents a weighted average position for the distribution.

Answer 1

The Correct Option is A

Concept: The mean, mathematical expectation, or Expected Value (\( \mu \) or \( E(X) \)) of a discrete random variable tells us the long-run average outcome value. It is calculated by finding the sum of each possible outcome value multiplied by its individual probability: \[ \mu = E(X) = \sum x_i \cdot P(x_i) \]

Step 1: Set up the individual product terms from the columns.
Multiply each value of variable \( X \) by its corresponding probability weight entry: \[ \mu = (1 \cdot P(1)) + (2 \cdot P(2)) + (3 \cdot P(3)) \] Substitute the numerical decimals from our table: \[ \mu = (1 \cdot 0.2) + (2 \cdot 0.5) + (3 \cdot 0.3) \]

Step 2: Calculate the products and sum them up to find the final mean.
Evaluate each multiplication step: \[ 1 \cdot 0.2 = 0.2 \] \[ 2 \cdot 0.5 = 1.0 \] \[ 3 \cdot 0.3 = 0.9 \] Add the values together to complete the calculation: \[ \mu = 0.2 + 1.0 + 0.9 = 2.1 \]