Suppose the useful lifetime in years of a personal computer pc is exponentially distributed with parameter a025 a student
entering a fouryear undergraduate program inherits a twoyearold pc from his sister who just graduated find the probability

Question

Suppose the useful lifetime, in years, of a personal computer (PC) is exponentially distributed with parameter A-0.25. A student entering a four-year undergraduate program inherits a two-year-old PC from his sister who just graduated. Find the probability the useful life time of the PC the student inherited will last at least until the student graduates.

Joe Evans · Answer

To solve this problem, we need to understand the concept of the exponential distribution and how it applies to the lifetime of the PC.

Given that the lifetime $X$ of a personal computer is exponentially distributed with a rate parameter $\lambda = 0.25$ , we have:

f(x) = \lambda e^{-\lambda x}

The probability that the PC lasts at least until the student graduates (4 more years) is found by calculating the survival function. Since the PC is already 2 years old, we are interested in the probability that it lasts an additional 4 years (totaling a useful life of 6 years).

The survival function for an exponential distribution is given by:

S(x) = P(X > x) = e^{-\lambda x}

We want $P(X > 6 \mid X > 2)$ , which is the probability that it lasts more than 6 years given that it is already 2 years old. By the memoryless property of the exponential distribution, this conditional probability is:

P(X > 6 \mid X > 2) = P(X > 4) = e^{-\lambda \cdot 4}

Plugging in the given rate $\lambda = 0.25$ :

P(X > 4) = e^{-0.25 imes 4} = e^{-1} \approx 0.3679

Therefore, the probability that the PC will last at least until the student graduates is approximately 0.3679, or 36.79%.

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