Question

The cumulative distribution function of a continuous random variable \( X \) is given by \( F(X = x) = \dfrac{\sqrt{x}}{2} \). Then \( P(X>1) \) is

• A. \( \dfrac{1}{3} \)
• B. \( \dfrac{1}{\sqrt{2}} \)
• C. \( \dfrac{1}{2} \)
• D. \( \dfrac{1}{4} \)

Hint:

For any continuous random variable, always use \[ P(X>a) = 1 - F(a) \]

Answer 1

The Correct Option is C

Step 1: Use the definition of probability.
\[ P(X>1) = 1 - P(X \leq 1) \] Step 2: Use the given CDF.
\[ P(X \leq 1) = F(1) = \frac{\sqrt{1}}{2} = \frac{1}{2} \] Step 3: Compute the probability.
\[ P(X>1) = 1 - \frac{1}{2} = \frac{1}{2} \] Step 4: Conclusion.
Hence, \( P(X>1) = \dfrac{1}{2} \).