Mathematics, 19.03.2021 18:30 sabrinamarie391
Suppose that an appliance is constructed in such a way that it requires that n independent electronic components are all functioning. Assume that the lifespan of each of these, Tj, is an exponential random variable with parameter λj.
a. Let X be the random variable giving the lifespan of the appliance. Find the CDF and PDF for X.
b. Find the expected value of X.
c. Now find the median lifespan of the appliance (that is, the time t at which half of the appliances are likely to have broken and half to be working ).
Answers: 3
![T_j \sim exp ( \lambda j) \ \ \ j = 1 (1) n \\ \\ f_{Tj}(tj) = \lambda j e^{-\lambda j tj } , tj0 \\ \\ P\Big[ Tj x\Big] = \int \limits ^{\infty}_{x} \lambda j e^{-\lambda j tj}\ dtj \\ \\ = e^{-\lambda j x}, x 0 \\ \\ \\ \\ a) F_x (x) \\ \\ = P[X \le x ] \\ \\ = 1 - P[X x] \\ \\ = 1 - \pi \limits ^{n}_{j =1} \Big\{ P[T_j x ] \Big \}](/tpl/images/1206/9737/7e2e3.png)
![= \dfrac{1}{\sum \limits ^n_{j=1} \lambda j} \ \ \int \limits ^{\infty}_{o} \Big [(\sum \limits ^n_{j=1} \lambda j )x \Big] e ^{-\Big ( \sum \limits ^{n}_{j=1} \lambda j \Big)x} \ \ d \Big( x \sum \limits ^n_{j=1} \lambda j \Big)](/tpl/images/1206/9737/36a22.png)
![= \dfrac{1}{\sum \limits ^n_{j= 1} \lambda j} \int \llimits ^{\infty}_{o} t e^{-t} \ dt \ \ \ \ \text{\Big[By transformation }t =( \sum \limits ^n_{j=1} \lambda j )x \Big]](/tpl/images/1206/9737/a0149.png)
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Suppose that an appliance is constructed in such a way that it requires that n independent electroni...
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