Exercise 2 : Let A, / > 0 and X, Y two independent, r.v. with probability density functions : fx (x) = de- 1xxo and fy (x) = He #2 1,>0. 1) Express the cumulative distribution function (CDF) of X, x -> FX (x) = P(X x, Y > x) = e-(to), for every T E R.. 3) Give the CDF of For of the r.v. M := min(X; Y). 4) Deduce the density function of M. What is the distribution of M? Exercice 3 : Let X1, X2, . .., Xn i.i.d random variables with common Exp(1) distribution. Let Mn := min(X1, X2, . . . , Xn) and Sn := XitX2+ . .. + Xn. 1) Use Exercise 2 question 2), in order to deduce the distribution of nMn . 2) We recall that lim (1+ 2 )" = ez, Vz E C. We also recall that the characteristic function of X1 is given by $x (u) := E[elux] = 1 = 1 - iu' , UER. 2)a) Give the expression of s. (u) : = Een],uER. 2)b) Calculate limn-+ $s, (u). Deduce the limit in distribution of Sn 2)c) What is the limit in distribution of San ? Vn