The correct answer is $\boxed{\text{A}. 0.368}$.
The probability density function (PDF) of a continuous random variable $X$ is a function $f_X(x)$ that gives the probability that $X$ takes on a value between $x$ and $x + dx$, for an infinitesimally small value of $dx$. In this case, the PDF is given by
$$f_X(x) = e^{-x}, \quad 0 < x < \infty$$
The probability that $X$ is greater than 1 is given by
$$P(X > 1) = \int_1^\infty f_X(x) dx = \int_1^\infty e^{-x} dx = \left[ -e^{-x} \right]_1^\infty = 1 – e^{-1} \approx 0.368$$
Option A is the correct answer. Option B is incorrect because the probability that $X$ is greater than 1 is not equal to 0.5. Option C is incorrect because the probability that $X$ is greater than 1 is not equal to 0.632. Option D is incorrect because the probability that $X$ is greater than 1 is not equal to 1.