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3. Consider the probability density functions m) = ofme- \"ux > 0} and 9(3) = 0,,(1 + :r)'3l{x > 0}, where C; and Cg are xed constants. (a) Find Or and 09. [3 marks] (b) Dene, for positive constants s, t, Hf(S,t) = L: f(:r) (ix/Loo f(:r) dz, His, r) = I; ate) arr/Is\" gm arm. By nding explicit expressions for H;(s,t) and Hg(s,t), determine whether H J\"(5, t) and H9 (3, t) are increasing or decreasing functions in s for any xed constant t. [8 marks] (e) Let V be the lifespan of a man, and W be the time taken by two opposing extremists to end mutual hatred. Which of the two density functions, f or 9, would you use for modelling (i) the distribution of V? (ii) the distribution of W? Explain your choices with the help of the results obtained in (b). [9 marks]