Question 3.9: The lifetime of certain equipment is described by a r.v. X w......
The lifetime of certain equipment is described by a r.v. X whose distribution is Gamma with parameters \alpha=2 and \beta={\frac{1}{3}}, , so that the corresponding p.d.f. is: f(x)=9x e^{-{3}x},\;\mathrm{for}\;x\gt 0.~ Determine the expected lifetime, the variation around it, and the probability that the lifetime is at least 1 unit of time.
Since ~E X=\alpha\beta{\mathrm{~and~}}V\!\!ar(X)=\alpha\beta^{2},~ we have here: E X={\frac{2}{3}} and V\!\!ar(X)={\frac{2}{9}}. Also,
P(X\gt 1)=\int_{1}^{\infty}9x e^{-3x}d x=\frac{4}{e^{3}}\simeq0.199.
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