Question

A device has two electronic components. Let ?1T1 be the lifetime of Component 1, and suppose ?1T1 has the exponential distribution with mean 5 years. Let ?2T2 be the lifetime of Component 2, and suppose ?2T2 has the exponential distribution with mean 4 years.

Suppose ?1T1 and ?2T2 are independent of each other, and let ?=min(?1,?2)M=min(T1,T2) be the minimum of the two lifetimes. In other words, ?M is the first time one of the two components dies.

a) For each ?>0t>0, find ?(?>?).

[Hint: If the minimum has to be bigger than ?t, what does that tell you about each of the lifetimes?]

b) Use Part a to identify the distribution of ?. Provide its name and parameter (or parameters, if there are more than one).

c) Find the numerical value of ?(?).

Answer 1

A random variable X ~exp() has pdf: f(x)= df: F(x) = P(Xx)=

Given, two lifetimes, T1~exp(1/5) and T2~exp(1/4) and T1 independent of T2.

M=min(T1,T2)