Question

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.

Accepted Answer

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.

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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Key Concepts

Expected Value of a Continuous Random Variable

Exponential Distribution and Its PDF

Integration by Parts