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The time until failure of an electronic device has an
Chapter 6, Problem 100SE(choose chapter or problem)
The time until failure of an electronic device has an exponential distribution with mean 15 months. If a random sample of five such devices are tested, what is the probability that the first failure among the five devices occurs
a after 9 months?
b before 12 months?
Questions & Answers
QUESTION:
The time until failure of an electronic device has an exponential distribution with mean 15 months. If a random sample of five such devices are tested, what is the probability that the first failure among the five devices occurs
a after 9 months?
b before 12 months?
ANSWER:Step 1 of 3
Let \(X_{1}, X_{2}, \ldots X_{5}\) are independent exponential distribution with a mean of 15 months.
The probability density function of \(X_{i}\) is
\(f_{x}(x)=\left\{\begin{array}{ll}
\frac{1}{15} e^{-\frac{x}{15}} & ; x>0 \\
0 & ; \text { Otherwise }
\end{array}\right.\)
The CDF of the exponential distribution is
\(F_{X}(x)=1-e^{-\frac{x}{15}}\)
Let Consider \(X_{(1)}=\min \left\{x_{1}, x_{2}, \ldots, x_{5}\right\}\) denote the first failure length among the five devices.
Now the pdf of \(X_{(1)}\) is
\(\begin{aligned}
f_{x_{(1)}}(x) & =n[1-F(x)]^{n-1} f(x) \\
& =5\left[1-\left(1-e^{-\frac{x}{15}}\right)\right]^{5-1} \frac{1}{15} e^{-\frac{x}{15}} \\
& =\frac{5}{15} e^{-\frac{4 x}{15}} e^{-\frac{x}{15}} \\
& =\frac{1}{3} e^{-\frac{x}{3}}
\end{aligned}\)