Please provide suitable solutions to the questions below providing examples where necessary
Question 1:
You purchase a certain product. The manual states that the lifetime TT of the product, defined as the amount of time (in years) the product works properly until it breaks down, satisfies
P(T≥t)=e-t5, for all t≥0.P(T≥t)=e-t5, for all t≥0.
For example, the probability that the product lasts more than (or equal to) 22 years is P(T≥2)=e-25=0.6703P(T≥2)=e-25=0.6703. I purchase the product and use it for two years without any problems. What is the probability that it breaks down in the third year?
Question 2
Let C1,C2,?,CMC1,C2,?,CM be a partition of the sample space SS, and AA and BB be two events. Suppose we know that
AA and BB are conditionally independent given Ci, for all i∈{1,2,?,M}i∈{1,2,?,M};
BB is independent of all Ci's
Prove that AA and BB are independent.