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Official PYQ · 2021Set 1 · Q18Probability: Random Variables and DistributionsNAT1 mark

GATE CS 2021 Set 1 Q18 — Probability: Random Variables and Distributions

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The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 22. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to 22 decimal places) is ____________.
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Answer: 0.35 : 0.39

Let XX be the random variable representing the lifetime of the component. It is given that XX is exponentially distributed with parameter λ=2\lambda = 2.

The expected lifetime (mean) of an exponentially distributed random variable is given by:

Expected Lifetime E[X]=1λ=12=0.5\text{Expected Lifetime } E[X] = \frac{1}{\lambda} = \frac{1}{2} = 0.5

For an exponential distribution, the probability that the random variable exceeds a threshold xx is:

P(X>x)=e−λxfor x≥0P(X > x) = e^{-\lambda x} \quad \text{for } x \ge 0

We need to find the probability that the lifetime exceeds the expected lifetime, i.e., P(X>E[X])P(X > E[X]):

P(X>E[X])=P(X>1λ)=e−λ⋅1λ=e−1P(X > E[X]) = P\left(X > \frac{1}{\lambda}\right) = e^{-\lambda \cdot \frac{1}{\lambda}} = e^{-1}

Calculating the numerical value:

e−1≈12.71828≈0.3679e^{-1} \approx \frac{1}{2.71828} \approx 0.3679

Rounding to 22 decimal places, the probability is 0.37 (which lies within the official range of 0.350.35 to 0.390.39).

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