Question: 2. We consider the time X until the the failure of a machine component. Once the component fails it is immediately replaced. We model the

2. We consider the time X until the the failure of a machine component. Once the component fails it is immediately replaced. We model the X with the following density: fx (x) = Aexp[-Ax], x E [0, 00), for some A > 0. This is called the exponential distribution with rate A. (a) Show that fx(r) is a density. (b) Show that the CDF Fx(x) = 1 - exp[-Ax]. (c) Show that the moment generating function Mx(t) = >/(X -t) and state for what values of t it is defined. (d) Compute E(X ) and Var(X)

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