# Une introduction aux motifs (motifs purs, motifs mixtes, by Yves André By Yves André

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Xm) such that F(Xl, ... , xm) = I f:~ f(Zl, ... , Zm) dZ l ... dZm then f is called the density function for the dJ. F. (X E A) for every Borel subset A s;;; /R m. In particular if f is continuous at the point x = (Xl' ... (x k :::; X k :::; Xk + dXk, 1 :::; k :::; m) = f(Xl, ... , xm) dXl ... dx m. " F(x). ,xm ..... CO When F has a density function f so does each Fk, and their densities fk are given by fk(xk) = f:oo ... f:oo f(x) dX 1 ... dXk-l dXk+1 ... dxm· The random variables X k are independent if and only if F(x) = n Fk(Xk), rn k=l and when F has a density function f this condition becomes n fk(Xk)' rn f(x) = k=l 41 Multidimensional Distribution Functions We say that X has finite moments of order riff Rm IIxll r dFx(x) < 00.

Exercises 1. 6. In a city with 60,000 inhabitants, a mass x-ray survey is planned so as to detect all the people with tuberculosis. " Suppose that in the city there are 2000 persons with moderate attacks of tuberculosis. " Find the mean and variance of X. 2. (Parzen [45J) A man with n keys wants to open his door. He tries the keys independently and at random. Let N. be the number of trials required to open the door. Find EN. ) if (i) unsuccessful keys are not eliminated from further selections.

Assume without loss of generality that fl = O. ) Let 52 III. Limit Laws Sn y" = -. Then n cpyJu) = cpn(~), where cp is the common characteristic function of the Xns. l = 0, it follows from Property (P4 ) of characteristic functions that cp(u) = 1 + o(u). :! O. Since this weak limit turns out to be a constant,it follows that in fact we obtaln-convergence in probabil~ D ~ Central Limit Theorem. Let Xl' Xz, ... ,. :! N(O, 1), n = Var X n • PROOF. l = wise simply replace Xn with Xn -; cpyJu) ° and (J = 1.