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 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
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Use the total probability being 1 and the mean equation to find the unknown probabilities.
Updated On: Oct 1, 2026
- \(0.3\)
- \(0.4\)
- \(0.2\)
- \(0.5\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
Let \(P(X = 1) = a\). Then \(P(X = 3) = 2a\), \(P(X = 2) = 0.3\), and let \(P(X = 0) = b\).
Step 2: Use the mean.
\[ E(X) = 0\cdot b + 1\cdot a + 2(0.3) + 3(2a) = 7a + 0.6 = 1.3 \Rightarrow a = 0.1 \]
Step 3: Use the total.
\(b + a + 0.3 + 2a = 1\), so \(b = 1 - 0.3 - 3(0.1) = 0.4\).
Step 4: Check.
The probabilities are 0.4, 0.1, 0.3, 0.2, and they add to 1. The mean is \(0.1 + 0.6 + 0.6 = 1.3\).
Final Answer:
\(P(X = 0) = 0.4\), option (B). \[ \boxed{0.4} \]
Let \(P(X = 1) = a\). Then \(P(X = 3) = 2a\), \(P(X = 2) = 0.3\), and let \(P(X = 0) = b\).
Step 2: Use the mean.
\[ E(X) = 0\cdot b + 1\cdot a + 2(0.3) + 3(2a) = 7a + 0.6 = 1.3 \Rightarrow a = 0.1 \]
Step 3: Use the total.
\(b + a + 0.3 + 2a = 1\), so \(b = 1 - 0.3 - 3(0.1) = 0.4\).
Step 4: Check.
The probabilities are 0.4, 0.1, 0.3, 0.2, and they add to 1. The mean is \(0.1 + 0.6 + 0.6 = 1.3\).
Final Answer:
\(P(X = 0) = 0.4\), option (B). \[ \boxed{0.4} \]
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