Probability of x failing before y

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The discussion revolves around calculating the probability that component X fails before component Y, given their independent failure densities. The density function for X is f(x) = exp(-x) and for Y is f(y) = 2*exp(-2y). The user attempts to derive the probability by integrating the failure densities over time T, aiming to find Pr(X fails before Y). They express uncertainty in their calculations, particularly in setting up the integrals correctly. The thread highlights the need for clarification on the integration limits and the correct formulation of the probability expression.
tony3333
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Homework Statement



hi,
i have a problem and i really want you to help me with it.
we have X and Y that fail independently of each other.
density of X : f(x)=exp(-x), and density of Y : f(y)=2*exp(-2y) what is the probability that X component fails first?
(it should be a number)
thank you
 
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You need to show some work. What have you tried?
 
so, i think, i need the probability of X failing and Y not, at a time T.
T T T oo
so it is Pr=∫f(x)dx *(1-∫f(y)dy)=∫2*exp(-2t)dt * ∫2*exp(-2t)dt=
0 0 0 T
and here i am stuck...


(the limits of the first integral are (0,T), second (0,T), third (0,T), and forth (T, +oo)
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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