$f(x) = e^{-x}$ for $0 < x < \infty$ (and 0 elsewhere) is the p.d.f. of an **exponential** random variable. The probability $P[X > 1]$ is:
A$e$
B$1 - 1/e$
C$1 - e$
D$1/e$
Answer & Solution
Correct answer: D. $1/e$
$P[X > 1] = \int_1^\infty e^{-x}\,dx = [-e^{-x}]_1^\infty = 0 - (-e^{-1}) = 1/e$. (Equivalently, $P[X \leq 1] = 1 - 1/e$.)
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