Smartphones are produced by two different manufacturers A and B. The smartphones from manufacturer A have an expected lifetime of 3 years, while those from manufacturer B have an expected lifetime of only 1 year. Assume that in both cases, the lifetimes are governed by an exponential distribution. Let TA, TB denote the random variables associated with the smartphone lifetimes (in years) from manufacturers A and B, respectively. Moreover, assume that TAand TB are independent.
(a) Suppose that you only buy smartphones from manufacturer A : Let S be the random variable S = TA¹ + TA² + TA³, where TAᴷ are independent samples of TA. Calculate P(S>5) and the expectation E[S].