Question

Given the probability density function (p.d.f.) of the random variable X, \(f(x) = \frac{1}{2a}\), \(0 < x < 2a\), \(a > 0\)
\(= 0\), otherwise, then which of the following is correct ?

• A. \(P(X < \frac{a}{2}) = P(X > \frac{a}{2})\)
• B. \(P(X < \frac{a}{2}) < P(X > \frac{3a}{2})\)
• C. \(P(X < \frac{a}{2}) > P(X > \frac{3a}{2})\)
• D. \(P(X < \frac{a}{2}) = P(X > \frac{3a}{2})\)

Hint:

Compute each probability as area under the constant density.

Answer 1

The Correct Option is D

Step 1: Density:
\(f(x)=\dfrac1{2a}\) on \((0,2a)\), so probability over an interval equals its length divided by \(2a\).

Step 2: Compute the Probabilities:
\(P\left(X<\dfrac a2\right)=\dfrac{a/2}{2a}=\dfrac14\).
\(P\left(X>\dfrac a2\right)=\dfrac{2a-a/2}{2a}=\dfrac34\).
\(P\left(X>\dfrac{3a}2\right)=\dfrac{2a-3a/2}{2a}=\dfrac14\).

Step 3: Compare:
\(P(X<a/2)=P(X>3a/2)=\tfrac14\). So (D) is true.

Step 4: Check the Others:
(A) compares \(\tfrac14\) and \(\tfrac34\), which are not equal. (B) says \(\tfrac14<\tfrac14\), false. (C) says \(\tfrac14>\tfrac14\), false.

Final Answer:
The correct statement is option (D). \[ \boxed{\text{(D)}} \]