Finding the PDF from a Given CDF

The cumulative distribution function (CDF) is defined as:

\( F(x) = 1 – e^{-2x} \) for \( x \geq 0 \)

\( F(x) = 0 \) for \( x < 0 \)

To find the probability density function (PDF), we take the derivative of the CDF:

\( f(x) = 2e^{-2x} \) for \( x \geq 0 \)

\( f(x) = 0 \) for \( x < 0 \)

Conclusion: The PDF is \( f(x) = 2e^{-2x} \) for \( x \geq 0 \), and \( f(x) = 0 \) for \( x < 0 \). This function describes the density of the probability distribution for the given range of \( x \).