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GATE 2018 ME (ME2) – Question 37

Engineering Mathematics · Probability and Statistics: Mean, median, mode, standard deviation, random variables and distributions · 2 marks · Multiple choice

Independent exponentially distributed random variables $X_1,X_2$ have means 0.5 and 0.25. The distribution of $Y=\min(X_1,X_2)$ is ______.

  1. exponential with mean 1/6
  2. exponential with mean 2
  3. normal with mean 3/4
  4. normal with mean 1/6

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Correct answer: (A) exponential with mean 1/6

Explanation

The rates are $\lambda_1=1/0.5=2$ and $\lambda_2=1/0.25=4$. For t ≥ 0, independence gives $$P(Y>t)=P(X_1>t)P(X_2>t)=e^{-2t}e^{-4t}=e^{-6t}.$$ Therefore Y is exponential with mean **1/6, A**.