GATE CS 2021 Set 1 Q.28 (CS Q.18): Probability and Statistics

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

Your answer

Official answer and solution

Accepted answers: 0.35 to 0.39
From: Official GATE 2021 answer key

Solution

Determine the expected value of an exponential distribution and evaluate the survival function at that value.

Let $X$ be an exponentially distributed random variable with parameter $\lambda > 0$ . Its probability density function is $f(x) = \lambda e^{-\lambda x}$ for $x \ge 0$ , and its cumulative distribution function is $F(x) = 1 - e^{-\lambda x}$ .

The expected value of $X$ is:

E[X] = \frac{1}{\lambda}

The probability that $X$ exceeds its expected value is given by the survival function evaluated at $x = \frac{1}{\lambda}$ :

P(X > E[X]) = P\left(X > \frac{1}{\lambda}\right) = 1 - F\left(\frac{1}{\lambda}\right) = e^{-\lambda (1/\lambda)} = e^{-1}

Computing the numerical value:

e^{-1} = \frac{1}{2.71828} \approx 0.367879... \approx 0.37

Rounded to 2 decimal places, the value is 0.37 (falling comfortably within the official range of 0.35 to 0.39).

Final answer

0.37

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2021 Computer Science and Information Technology (CS, Set-1)
Question: 28 of 65, printed as CS Q.18
Section: Computer Science and Information Technology

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