GATE 2026 PI – Question 41

Quality and Reliability · Reliability and Maintenance · 2 marks · Multiple choice

The failure-rate function is $r(t)=f(t)/(1-F(t))$, where f is the failure-time density and $F(t)=1-e^{-at^b}$, a > 0, b > 0. Find r(t).

$abt^{b-1}e^{-at^b}$
$abt^{b-1}$
$1-abt^{b-1}$
$1-abt^{b-1}e^{-at^b}$

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Show answer and explanation

Correct answer: (B) $abt^{b-1}$

Explanation

Differentiate the CDF: $f(t)=abt^{b-1}e^{-at^b}$. Survival probability is $1-F(t)=e^{-at^b}$. Dividing cancels the exponential, leaving **$r(t)=abt^{b-1}$, B**. A hazard rate is conditional on survival to t.

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