If the probability function for a random variable \( x \) is given as \( f(x) = \frac{x+3}{15} \) when \( x = 1, 2, 3 \), find the sum of the values of the probability distribution for \( x \).
Step 1: Verify the sum of probabilities.
For a discrete random variable, the sum of all probabilities must equal 1. The given probability mass function is:\[f(x) = \frac{x+3}{15} \text{for } x = 1, 2, 3.\]We must confirm that the sum of these probabilities equals 1.
Step 2: Compute individual probabilities.
- For \( x = 1 \), \( f(1) = \frac{1+3}{15} = \frac{4}{15} \) - For \( x = 2 \), \( f(2) = \frac{2+3}{15} = \frac{5}{15} \) - For \( x = 3 \), \( f(3) = \frac{3+3}{15} = \frac{6}{15} \)
Step 3: Sum the probabilities.
The sum of the calculated probabilities is:\[\frac{4}{15} + \frac{5}{15} + \frac{6}{15} = \frac{15}{15} = 1.\]
Step 4: Final determination.
As the sum of probabilities is 1, the correct option is (D).