A continuous random variable has normalized density $f(x)=e^{-a|x|}$ for all real x, where a is a real constant. Find its variance to one decimal place.
Normalization requires a>0 and ∫e⁻ᵃ|ˣ|dx=2/a=1, so a=2. Symmetry gives mean zero. The second moment is 2∫₀∞x²e⁻²ˣdx=2×2!/2³=.5. Therefore variance=.5−0²=.5. Treating a as arbitrary would ignore the missing normalizing prefactor.
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