The lifetime of a component of a certain type is a random variable whose probability density function is exponentially d...

Question

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 ___________

Accepted Answer

0.37 (i.e., e^{-1} ≈ 0.3679). An exponential random variable X with rate parameter lambda = 2 has expected value E[X] = 1/lambda = 0.5. We need P(X > E[X]) = P(X > 1/lambda). For any exponential distribution this equals e^{-lambda * (1/lambda)} = e^{-1} regardless of lambda. So the answer is e^{-1} ≈ 0.3679, rounded to 0.37.