Question

A machine needs 4 out of its 6 identical independent components to operate. Let X1, X2, ..., X. denote the respective lifetimes of the components, and assume that each component's lifetime is exponentially distributed with a mean of 1/λ hours. Find:

(a) The CDF of the machine's lifetime.

(b) The PDF of the machine's lifetime.

Answer 1

Solution:

Let X denotes the lifetime of the machine.

P(X t) = 1 - P(X > t)

P(X > t) = P(at least 4 out of its 6 identical components have lifetime >t)

P(i out of 6 identicl components has lifetime > t)