Question:
Let \(F(x)\) be the cumulative distribution function (c.d.f.) of a continuous random variable \(X\). If \(F(b) = 0.7\) and \(P(X > a) = 0.4\), then the value of \(P(a < X < b)\) is ...
Let \(F(x)\) be the cumulative distribution function (c.d.f.) of a continuous random variable \(X\). If \(F(b) = 0.7\) and \(P(X > a) = 0.4\), then the value of \(P(a < X < b)\) is ...
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For a continuous variable, P(a < X < b) = F(b) - F(a).
Updated On: Oct 1, 2026
- \(0.1\)
- \(0.2\)
- \(0.3\)
- \(0.5\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
The c.d.f. is \(F(x) = P(X \leq x)\). For a continuous variable, single points have zero probability, so \(P(X > a) = 1 - F(a)\).
Step 2: Key Formula or Approach:
\(P(a < X < b) = F(b) - F(a)\).
Step 3: Detailed Explanation:
\(P(X > a) = 0.4\), so \(F(a) = 1 - 0.4 = 0.6\).
\[ P(a < X < b) = F(b) - F(a) = 0.7 - 0.6 = 0.1 \]
For a continuous variable \(P(X = a)\) and \(P(X = b)\) are zero, so using strict or non-strict inequality does not change the answer. The other options (0.2, 0.3, 0.5) would need \(F(a)\) to be \(0.5\), \(0.4\) or \(0.2\).
Final Answer:
\(P(a < X < b) = 0.1\), option (A). \[ \boxed{0.1} \]
The c.d.f. is \(F(x) = P(X \leq x)\). For a continuous variable, single points have zero probability, so \(P(X > a) = 1 - F(a)\).
Step 2: Key Formula or Approach:
\(P(a < X < b) = F(b) - F(a)\).
Step 3: Detailed Explanation:
\(P(X > a) = 0.4\), so \(F(a) = 1 - 0.4 = 0.6\).
\[ P(a < X < b) = F(b) - F(a) = 0.7 - 0.6 = 0.1 \]
For a continuous variable \(P(X = a)\) and \(P(X = b)\) are zero, so using strict or non-strict inequality does not change the answer. The other options (0.2, 0.3, 0.5) would need \(F(a)\) to be \(0.5\), \(0.4\) or \(0.2\).
Final Answer:
\(P(a < X < b) = 0.1\), option (A). \[ \boxed{0.1} \]
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