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Let X be the lifespan of an electrical component in days. X is a random variable that has an exponential distribution with parameter $\frac{1}{12}$. All component lifespans are independent.

let Y be the lifespan of an electrical device, which consists of 3 electrical components. All of the components must work properly in order for the device to operate.

What is the Cumulative distribution function of Y?

First, I think I need to define $Y$ as a equation. Is it $X$? $X^3$? $3X$? How can I have the intonation for that?

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The cdf of $Y$ is $$\begin{align*}F_Y(y)&=Pr(Y\leq y)\\ &=Pr(\text{At least one component has failed in less than y days})\\ &=1-Pr(\text{No components have failed in y days})\end{align*}.$$ The probability that no components have failed in y days should be easier to calculate, because you only have to deal with one case, whereas to find the probability that at least one component has failed in y days, you need to think about the probability that component 1 fails and the other 2 work, the probability that component 1 and 2 fail, etc.

We know the probability that each component hasn't failed in $y$ days is $1-F(y)$, where $F$ is the cdf for the exponential distribution with parameter $\frac{1}{12}$. So $$F_y(Y)=1-(1-F(y))^3=1-(e^{-\frac{1}{12}y})^3=1-e^{-\frac{1}{4}y}$$ So $Y$ is also exponentially distributed, with parameter $1/4$.

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