Proving the Integral Relationship for e^(-2x)

This question asks the candidate to verify that a particular improper integral of an exponential decay function equals one. The setup is the standard exponential density defined on the positive real line with a fixed rate parameter, and the task is to show that its total area is finite and normalized. In probability terms, the integrand corresponds to a probability density function, and the integral represents the total probability over all nonnegative values of the variable. Such checks of normalization commonly appear in introductory quant, data science, and statistics interviews when continuous distributions are first introduced.

To answer it cleanly, the candidate must set up and evaluate an improper integral using a basic antiderivative, then handle the limit as the upper bound goes to infinity. It leans on comfort with exponential functions, elementary integration, and convergence of tails of decaying exponentials. Interviewers look for correct treatment of the infinite limit, clear justification that the boundary term vanishes, and an understanding that the result confirms the function is a valid probability density.

What it tests

Integrals of the form $\int_0^\infty a e^{-a x} dx$ represent the total probability for an exponential distribution with rate parameter $a$. The key structure is that the exponential function decays rapidly enough as $x \to \infty$ to ensure convergence, and the normalization constant $a$ ensures the area under the curve is exactly 1. This is a general property: for any $a > 0$, $\int_0^\infty a e^{-a x} dx = 1$. The reason is that the antiderivative $-e^{-a x}$ evaluated from 0 to $\infty$ always yields 1, reflecting the fact that the exponential distribution is a probability density function. The underlying pattern is that the integral of a properly normalized exponential decay over its support always sums to unity, regardless of the rate parameter.