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GATE CSE 2021 Set 1 Q28: Exponential lifetime exceeding its mean

Question

The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to 2 decimal places) is ________.

Try it: numerical answer: type the value

Numerical answer. No negative marking.
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Answer: 0.35 to 0.39 (official GATE 2021 final answer key)

Concept used: For an exponential variable, P(X>x)=e−λxP(X > x) = e^{-\lambda x} and E[X]=1/λE[X] = 1/\lambda.

First step: P(X>E[X])=e−λ⋅(1/λ)P(X > E[X]) = e^{-\lambda \cdot (1/\lambda)}, which does not depend on the parameter.

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Source: official GATE 2021 Set 1 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key

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