The lifetime of a component of a certain type is a random variable whose…
2021
The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to 2 decimal places) is ____________.
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Correct answer: 0.35 to 0.39
Key insight: for an exponential distribution with rate parameter 2, the expected lifetime (mean) is 1/2 = 0.5.
Compute the probability that the lifetime exceeds the mean:
For an exponential random variable with rate 2, P(X > x) = e^{-2x}.
Evaluate at x = mean = 0.5: P(X > 0.5) = e^{-2 * 0.5} = e^{-1} ≈ 0.367879...
Rounded to two decimal places, the probability is 0.37.
Answer: 0.37