Question

The p.d.f. of a continuous random variable \( X \) is given by \[ f(x) = \frac{x + 2}{18}, \quad \text{if} \, -2<x<4, \quad f(x) = 0, \, \text{otherwise}. \] Then \( P[ |x|<1 ] = \)

• A. \( \frac{1}{18} \)
• B. \( \frac{4}{9} \)
• C. \( \frac{2}{9} \)
• D. \( \frac{1}{9} \)

Hint:

To compute probabilities for continuous random variables, always integrate the p.d.f. over the desired interval.

Answer 1

The Correct Option is C

Step 1: Setting up the probability.
We are asked to find \( P[ |x|<1 ] \), which means we need to integrate the probability density function \( f(x) \) over the interval \( -1<x<1 \). The formula for the probability is: \[ P[ |x|<1 ] = \int_{-1}^{1} f(x) \, dx \] Since \( f(x) = \frac{x + 2}{18} \), we substitute this into the integral.
Step 2: Solving the integral.
We calculate the integral: \[ P[ |x|<1 ] = \int_{-1}^{1} \frac{x + 2}{18} \, dx \] After solving the integral, we obtain the value \( P[ |x|<1 ] = \frac{2}{9} \).
Step 3: Conclusion.
Thus, the probability is \( \frac{2}{9} \), which makes option (C) the correct answer.