Question

The probability density function of a random variable \(X\) is given by \[ f(x) = \frac{k}{\sqrt{x}}, \quad 0 \le x \le 4 \] \[ f(x) = 0, \quad \text{otherwise} \] Find \(P(1<X<4)\).

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

Hint:

Always determine the normalizing constant of a probability density function before calculating probabilities.

Answer 1

The Correct Option is A

Step 1: Find the value of constant \(k\).
Since \(f(x)\) is a probability density function, \[ \int_{0}^{4} \frac{k}{\sqrt{x}} \, dx = 1 \]
Step 2: Evaluate the integral.
\[ k \int_{0}^{4} x^{-1/2} dx = k \left[ 2\sqrt{x} \right]_{0}^{4} \] \[ = k (2 \times 2) = 4k \]
Step 3: Solve for \(k\).
\[ 4k = 1 \Rightarrow k = \frac{1}{4} \]
Step 4: Find \(P(1<X<4)\).
\[ P(1<X<4) = \int_{1}^{4} \frac{1}{4\sqrt{x}} dx \] \[ = \frac{1}{4} \left[ 2\sqrt{x} \right]_{1}^{4} \] \[ = \frac{1}{4} (4 - 2) = \frac{1}{2} \]