Question

Let X be a discrete random variable. The probability distribution of X is given below

and E(X) = 4, then the value of AB is equal to

• A. \( \frac{3}{10} \)
• B. \( \frac{2}{15} \)
• C. \( \frac{1}{15} \)
• D. \( \frac{3}{20} \)

Hint:

Use the sum of probabilities and the given mean to find the two unknown probabilities, then multiply them.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The distribution has X = 30, 10, -10 with probabilities \(\frac{1}{5}\), A, B. The probabilities add up to 1 and the mean is \(E(X)=\sum xP(x)\).

Step 2: Use the Sum of Probabilities
\(\frac{1}{5}+A+B=1\), so \(A+B=\frac{4}{5}\).

Step 3: Use the Mean
\(E(X)=30\cdot\frac{1}{5}+10A-10B=4\), so \(6+10(A-B)=4\) and \(A-B=-\frac{1}{5}\).

Step 4: Solve for A and B
Adding the two relations: \(2A=\frac{3}{5}\), so \(A=\frac{3}{10}\) and \(B=\frac{4}{5}-\frac{3}{10}=\frac{1}{2}\).

Step 5: Compare the options
\(AB=\frac{3}{10}\times\frac{1}{2}=\frac{3}{20}\), which matches the option \(\frac{3}{20}\).

Final Answer:
The value of AB is \(\frac{3}{20}\). \[ \boxed{\text{(D) }\dfrac{3}{20}} \]