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: Symmetry:
The uniform density on $(0,2a)$ is symmetric about $x=a$. The interval $(0,a/2)$ and the interval $(3a/2,2a)$ are mirror images of each other about $x=a$ and have equal length $a/2$.

Step 2: Conclusion:
Equal lengths under a constant density give equal probabilities, so $P(X<a/2)=P(X>3a/2)$.

Step 3: Reject Others:
The event $X>a/2$ has length $3a/2$, three times the length of $X<a/2$, so (A) is false. Options (B) and (C) assert strict inequality between equal quantities. Option (D).

Final Answer:
Option (D). \[ \boxed{\text{(D)}} \]