A device contains three components, each of which has time in hours with the PDF $f(x)= \begin{cases} \frac{2x}{10^2} e^{-(x/10)^2}, & 0<x<\infty \\ 0, & \text{elsewhere} \end{cases}$ The devcie with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour or its operation?
We can let Y=min{$X_1,X_2,X_3$}, then we take the integral of f(x), and what's next? I want the general idea of approaching. (Some helpful links for sample problems are appreciated too)
Let $X_i$ be the components of the device distributed according to the density $f$. Then the device does not fail if all three components last more than an hour i.e. $X_1>1$, $X_2>1$ and $X_3>1$. Hence probability of failure in the first hour is given by $$ P(\text{failure})=1-P(X_1>1, X_2>1, X_3>1)=1-P(X_1>1)^3 $$ where we have used the i.i.d assumption. You can compute $P(X_1>1)=1-P(X_1\leq 1)$ by integrating the density.