Question

A random variable X takes the values 0, 1, 2, 3 and its mean is \(1.3\). If \(P(X = 3) = 2P(X = 1)\) and \(P(X = 2) = 0.3\), then \(P(X = 0)\) is

• A. \(0.3\)
• B. \(0.4\)
• C. \(0.2\)
• D. \(0.5\)

Hint:

Use the total probability being 1 and the mean equation to find the unknown probabilities.

Answer 1

The Correct Option is B

Step 1: Unknowns.
Let $p_0, p_1, p_2, p_3$ be the four probabilities. Given $p_2 = 0.3$ and $p_3 = 2p_1$.

Step 2: Two equations.
Sum: $p_0 + p_1 + 0.3 + 2p_1 = 1$, so $p_0 + 3p_1 = 0.7$. Mean: $p_1 + 0.6 + 6p_1 = 1.3$, so $p_1 = 0.1$.

Step 3: Solve.
$p_0 = 0.7 - 0.3 = 0.4$.

Final Answer:
Option (B). \[ \boxed{0.4} \]