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Citrus2011 [14]
2 years ago
8

The lifetime of two light bulbs are modeled as independent and exponential random variables X and Y, with parameters lambda and

mu, respectively. The time at which one of those two light bulb first burns out is Z = min (X, Y), what is the PDF of Z?
Mathematics
1 answer:
Scorpion4ik [409]2 years ago
7 0

Answer:

\bf h(x)=max(\lambda,\mu) e^{-max(\lambda,\mu) x}\;(x\geq0)

Step-by-step explanation:

The PDF of X is

\bf f(x)=\lambda e^{-\lambda x}\;(x\geq0)

The PDF of Y is

\bf g(x)=\mu e^{-\mu x}\;(x\geq0)

The means of X and Y are respectively,

\bf \displaystyle\frac{1}{\lambda}\;,\displaystyle\frac{1}{\mu}

so we can see that the larger the parameter, the smaller the mean. Hence the PDF of Z = min(X, Y) is an exponential with the largest parameter of the two.

Therefore, the PDF of Z is

\bf h(x)=max(\lambda,\mu) e^{-max(\lambda,\mu) x}\;(x\geq0)

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