The GATE Grind

GATE 2025 DA – Question 21

Probability and Statistics · Random variables, discrete and continuous distributions · 1 mark · Multiple choice

It is given that $P(X \ge 2) = 0.25$ for an exponentially distributed random variable $X$ with $E[X] = \frac{1}{\lambda}$, where $E[X]$ denotes the expectation of $X$. What is the value of $\lambda$? ($\ln$ denotes natural logarithm)

  1. $\ln 2$
  2. $\ln 4$
  3. $\ln 3$
  4. $\ln 0.25$

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Show answer and explanation

Correct answer: (A) $\ln 2$

Explanation

For an exponential distribution $P(X \ge x) = e^{-\lambda x}$, so $e^{-2\lambda} = 0.25 = \frac{1}{4}$. Then $2\lambda = \ln 4 = 2\ln 2$, so $\lambda = \ln 2$.