63. Average Lifetime The average time until a computer chip fails (see Exercise 62) is 0.00005 ∫(from 0 to ∞) t e^(-0.00005t) dt. Find this value.

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Recognize that the problem asks for the average lifetime, which is given by the integral \(0.00005 \int_0^{\infty} t e^{-0.00005 t} \, dt\). This is an expected value calculation for a continuous random variable with probability density function proportional to \(e^{-\lambda t}\), where \(\lambda = 0.00005\).

Recall the formula for the expected value of an exponential distribution: if \(X\) has PDF \(f(t) = \lambda e^{-\lambda t}\) for \(t \geq 0\), then \(E[X] = \int_0^{\infty} t \lambda e^{-\lambda t} \, dt = \frac{1}{\lambda}\).

Identify that the integral inside the problem matches the form \(\int_0^{\infty} t e^{-\lambda t} \, dt\), but the problem includes the factor \(0.00005\) outside the integral, which corresponds to \(\lambda\).

Use integration by parts to verify the integral \(\int_0^{\infty} t e^{-\lambda t} \, dt\): let \(u = t\) and \(dv = e^{-\lambda t} dt\), then compute \(du\) and \(v\), and apply the integration by parts formula \(\int u \, dv = uv - \int v \, du\).

After evaluating the integral, multiply the result by \(0.00005\) as given, and simplify to find the average lifetime value.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Expected Value of a Continuous Random Variable

The expected value (or mean) of a continuous random variable is calculated as the integral of the variable multiplied by its probability density function over its entire range. It represents the average or mean outcome one would expect over many trials.

Exponential Distribution and Its PDF

The exponential distribution models the time between events in a Poisson process and has a probability density function (PDF) of the form λe^(-λt) for t ≥ 0. It is often used to model lifetimes or waiting times, where λ is the rate parameter.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the formula ∫u dv = uv - ∫v du and is useful for solving integrals involving polynomials multiplied by exponentials, such as t e^(-at).