Question

The p.d.f. of a continuous r.v. \( X \) is given by \[ f(x) = \frac{x}{8}, \quad 0<x<4 \quad \text{and} \quad f(x) = 0, \text{ otherwise,} \] then \( P(X \leq 2) \) is

• A. \( \frac{5}{16} \)
• B. \( \frac{9}{16} \)
• C. \( \frac{7}{16} \)
• D. \( \frac{1}{4} \)

Hint:

To find probabilities from a p.d.f., integrate the p.d.f. over the desired range of values.

Answer 1

The Correct Option is D

Step 1: Write the cumulative distribution function.
The cumulative distribution function (CDF) is the integral of the probability density function (PDF) from \( 0 \) to \( x \): \[ P(X \leq 2) = \int_0^2 \frac{x}{8} dx. \]
Step 2: Perform the integration.
Perform the integration: \[ P(X \leq 2) = \left[ \frac{x^2}{16} \right]_0^2 = \frac{4}{16} = \frac{1}{4}. \]
Step 3: Conclusion.
Thus, the correct answer is \( \frac{1}{4} \), corresponding to option (D).