The probability that the lifetime of at least one component exceeds 2 is 0.135.
Given that the joint pdf is f(x,y)=xe^(-x(1+y)) for x≥0, y≥0 and 0 otherwise as shown in attached image.
We want to find the probability that the lifetime of at least one component exceeds 2.
The probability that the lifetime of at least one component exceeds 2 is P(X>2).
[tex]\begin{aligned}P(X > 2)&=\int_{x=2}^{\infty}\int_{y=0}^{\infty}f(x,y)dydx\end[/tex]
Now, we will substitute the given function, we get
[tex]\begin{aligned}P(X > 2)&=\int_{x=2}^{\infty}\int_{y=0}^{\infty}xe^{-x(1+y)}dydx\end[/tex]
Further, we will simplify this, we get
[tex]\begin{aligned}P(X > 2)&=\int_{x=2}^{\infty}\left[-e^{-x(1+y)\right]_{0}^{\infty}dx\\ &=\int_{x=2}^{\infty}e^{-x}dx\\ &=\left[-e^{-x}\right]_{2}^{\infty}\\ &=e^{-2}\\ &=0.135\end[/tex]
Hence, the probability that the lifetime of at least one component exceeds 2 for the joint pdf is f(x,y)=xe^(-x(1+y)) for x≥0, y≥0 and 0 otherwise as shown in attached image is 0.135.
Learn more about the joint pdf from here brainly.com/question/15109814
#SPJ4
