Question

Let X be a continuous random variable with the probability density function(p.d.f.) given by
\(f(x) = \left\{ \begin{array}{cc}kx, & 0\leq x < 1 \\ k, & 1\leq x < 2 \\ -kx+3k, & 2\leq x < 3 \\ 0, & \text{otherwise}\end{array} \right.\)
\(P(2 < X\leq 3) = \cdots\)

• A. \(\frac{1}{2}\)
• B. \(\frac{1}{3}\)
• C. \(\frac{1}{4}\)
• D. \(\frac{1}{5}\)

Hint:

Total probability is 1, which fixes k; then integrate the last piece.

Answer 1

The Correct Option is C

Step 1: Use geometry:
The pdf is a trapezoid-like shape: it rises from 0 to $k$ on $[0,1]$, stays at $k$ on $[1,2]$, and falls from $k$ to 0 on $[2,3]$.

Step 2: Areas:
Left triangle: $\frac12\cdot1\cdot k$. Middle rectangle: $1\cdot k$. Right triangle: $\frac12\cdot1\cdot k$. Total $2k = 1$, so $k = \frac12$.

Step 3: Answer:
$P(2 < X \le 3)$ is the right triangle, with area $\frac k2 = \frac14$.

Final Answer:
The probability is 1/4, option (C). \[ \boxed{\frac{1}{4}} \]