Question

If the probability density function of a continuous random variable is \[ f(x) = \frac{x^3}{3} \quad \text{if} \quad -1<x<2, \quad f(x) = 0 \quad \text{otherwise}, \] then the cumulative distribution function of \( X \) is

• A. \( \frac{1}{14} \left( x^4 - 1 \right) \)
• B. \( \frac{1}{10} \left( x^4 - 1 \right) \)
• C. \( \frac{1}{16} \left( x^4 - 1 \right) \)
• D. \( \frac{1}{12} \left( x^4 - 1 \right) \)

Hint:

To find the CDF from a given PDF, integrate the PDF over the desired range of \( x \).

Answer 1

The Correct Option is D

Step 1: Formula for the cumulative distribution function (CDF).
The cumulative distribution function (CDF) is obtained by integrating the probability density function (PDF) from the lower limit of \( x \) to the variable \( x \). So, we integrate \( f(x) \): \[ F(x) = \int_{-1}^{x} \frac{t^3}{3} \, dt \]
Step 2: Perform the integration.
We compute the integral: \[ F(x) = \frac{1}{3} \int_{-1}^{x} t^3 \, dt = \frac{1}{3} \cdot \frac{x^4}{4} - \frac{1}{3} \cdot \frac{(-1)^4}{4} = \frac{1}{12} \left( x^4 - 1 \right) \]
Step 3: Conclusion.
Thus, the cumulative distribution function is \( F(x) = \frac{1}{12} \left( x^4 - 1 \right) \), corresponding to option (D).