Step 1: Understanding the Concept:
For any discrete probability distribution, two fundamental properties must hold:
1. The sum of all probabilities must equal 1: \( \sum P(X=x) = 1 \).
2. The expected value (mean) is the sum of the products of each value and its probability: \( E(X) = \sum [x \cdot P(X=x)] \). Step 2: Key Formula or Approach:
We have two unknowns, \( A \) and \( B \). We can set up a system of two linear equations using the two properties mentioned above.
Equation 1: \( \frac{1}{5} + A + B = 1 \)
Equation 2: \( 30(\frac{1}{5}) + 10(A) + (-10)(B) = E(X) \) Step 3: Detailed Explanation:
From the property that sum of probabilities is 1:
\[ \frac{1}{5} + A + B = 1 \]
\[ A + B = 1 - \frac{1}{5} \]
\[ A + B = \frac{4}{5} \quad \dots \text{(Equation 1)} \]
We are given \( E(X) = 4 \). Let's calculate the expected value:
\[ E(X) = (30 \times \frac{1}{5}) + (10 \times A) + (-10 \times B) \]
\[ 4 = 6 + 10A - 10B \]
Divide the entire equation by 2 to simplify:
\[ 2 = 3 + 5A - 5B \]
\[ 5A - 5B = -1 \]
Divide by 5:
\[ A - B = -\frac{1}{5} \quad \dots \text{(Equation 2)} \]
Now, solve the system of linear equations:
(1) \( A + B = \frac{4}{5} \)
(2) \( A - B = -\frac{1}{5} \)
Add (1) and (2):
\[ 2A = \frac{4}{5} - \frac{1}{5} = \frac{3}{5} \]
\[ A = \frac{3}{10} \]
Subtract (2) from (1):
\[ 2B = \frac{4}{5} - \left(-\frac{1}{5}\right) = \frac{5}{5} = 1 \]
\[ B = \frac{1}{2} \]
We need to find the value of \( AB \):
\[ AB = \left(\frac{3}{10}\right) \left(\frac{1}{2}\right) = \frac{3}{20} \]
Step 4: Final Answer:
The value of \( AB \) is \( \frac{3}{20} \).
