GATE 2014 EC (EC2) – Question 59
A one-bit quantizer input X has density $f_X(x)=2e^{-2x}$ for $x\ge0$, and zero otherwise. Find the threshold making the two outputs equiprobable.
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Correct answer: 0.34 to 0.36
Explanation
The density2e⁻²ˣ for x≥0 has cumulative distribution1−e⁻²ˣ. Equal output probabilities require P(X≤a)=1/2, hence e⁻²ᵃ=1/2 and a=(ln2)/2=**0.346574**. This is the median rather than the distribution’s mean0.5.