Two signals are transmitted through a processor. The transmission times are independent random variables X₁ and X₂ with exponential distributions with respective parameters ₁ > 0 and ₂ > 0, with ₁ < ₂. It takes the processor an amount T of time to transmit both signals. Find the density function f_T (t) of the variable T, its expected value E(T), its variance Var(T), and its hazard rate function r_T (t), and compute limt r_T (t) in the following two cases:

(a) the signals are transmitted sequentially, that is the second signal can be transmitted only after the first signal has been transmitted (i.e. T=X₁+X₂);

(b) the signals are transmitted simultaneously (i.e. T = max(X₁, X₂))